I 


RATIONAL  ARITHMETIC 


COMPLETE 


BY 

GEORGE   P.    LORD 

n 


•  ■>_  >  ;« J 


>      O  5     .    ,  »     °     I     '      ' 


THE  GREGG  PUBLISHING  COMPANY 

NEW  YORK  CHICAGO  BOSTON  SAN  FRANCISCO 

LIVERPOOL 


1 


.copyright,  192  0,  by  the 
'•/:.•':  *:geegg  publishing  company 


'    •'. 


1    • 


•     •»  •    • 


A5a 


PREFACE 

Rational  Arithmetic  is  intended  for  use  in  business  colleges^ 
and  in  commercial  high  schools,  by  pupils  who  have  com- 
pleted the  equivalent  of  the  eighth  or  ninth  grade  in  the 
public  school  system. 

While  deficiencies  of  early  training  may  be  remedied  by 
its  use,  it  is  not  intended  as  a  textbook  for  those  who  are 
approaching  the  subject  for  the  first  time.  Neither  is  it 
intended  to  take  the  place  of  any  of  the  many  excellent 
works  now  in  use  in  the  grades  for  the  purpose  of  develop- 
ing a  general  understanding  of  mathematical  principles. 
Such  books,  while  they  have  satisfactorily  discharged  this 
function,  have  failed  to  develop  the  accuracy  and  facility 
so  vitally  essential  in  commercial  calculations. 

Other  commercial  arithmetics  have  tried  to  overcome  this 
weakness  by  following  similar  plans  of  instruction  in  abridged 
form.  Rational  Arithmetic  follows  a  very  different  plan. 
It  is  purely  a  vocational  work  and  aims  to  teach  the  "how'* 
rather  than  the  "why."  It  is  a  reference  book  of  com- 
mercial operations,  rather  than  a  method  of  presentation, 
and  should  be  so  used. 

Part  One  is  a  collection  of  practice  exercises  arranged 
along  the  lines  of  the  generally  accepted  order  of  presentation. 

Part  Two  contains  illustrated  solutions  covering  the  entire 
range  of  commercial  arithmetic  as  generally  understood. 
The  methods  used  are  those  of  business.  The  explanations 
are  expressed  in  language  which  may  be  understood  easily, 
rather  than  in  the  more  scholarly  language  usually  employed. 

iii 

460956 


iv  PREFACE 

References  throughout  the  book  are  by  paragraph  num- 
bers, which  will  allow  the  pupil  to  ascertain  for  himself  the 
best  method  of  solving  any  desired  problem. 

The  aim  has  been  to  produce  a  book  so  elastic  that  the 
teacher  may  arrange  a  course  of  study  to  suit  himself.  The 
author  has  found  it  advisable,  however,  to  start  pupils  with 
the  subject  of  balancing  accounts  which  arouses  their  in- 
terest and  gives  them  something  new  and  practical. 

Drill  on  decimals  should  immediately  follow  this,  for  the 
purpose  of  developing  accuracy  in  locating  the  decimal  point. 

The  advisability  of  work  on  the  subject  of  fractions  de- 
pends entirely  upon  the  attainments  of  the  individual  pupil. 
The  writer  has  found  that  fully  75  per  cent  of  his  pupils  are 
greatly  benefited  by  taking  up  this  subject  before  beginning 
the  strictly  commercial  work  which  commences  with  the 
subject  of  aliquot  parts. 

It  is  suggested  that  the  problems  in  addition  at  the  be- 
ginning of  Part  One  be  used  as  drill  problems  throughout  the 
course. 

Teachers  will  not  find  it  necessary  to  use  all  the  problems 
provided  for  each  subject.  The  aim  has  been  to  give  enough 
problems  to  meet  any  demand  that  may  arise. 

No  claim  is  made  for  originality  in  any  of  the  methods 
presented.  Every  method  that  appears  in  this  book  may 
be  found,  in  some  form,  elsewhere.  To  give  credit  to  the 
sources  from  which  the  author  has  obtained  assistance  in 
the  compilation  of  this  book  would  be  to  name  all  the  text- 
books consulted  by  him  in  an  experience  of  nearly  thirty 
years  as  arithmetic  teacher. 

George  P.  Lord 

Salem,  Mass. 


CONTENTS 
PART  ONE 

PAGE 

Preliminary  Problems 1 

Addition 1 

Subtraction ^ 

Decimals 1^ 

Multiplication 1^ 

Division   ....••••••  1^ 

Fractions ^^ 

Addition  of  Fractions 16 

Subtraction  of  Fractions 17 

Multiplication  of  Fractions 17 

Division  of  Fractions 18 

Practice  Problems  —  Fractions  and  Decimals  .         .         .18 

Denominate  Numbers 2^ 

Aliquot  Parts ^"^ 

Exercises  in  Billing 30 

Percentage  ....••••••  ^^ 

General  Problems  in  Percentage 38 

Profit  and  Loss  ....•••••  "^^ 
General  Problems  in  Profit  and  Loss         .         .         .         .47 

Trade  Discount ^^ 

General  Problems  in  Trade  Discount        .         .         .         .55 

Commission ^^ 

General  Problems  in  Commission 66 

Time 69 

V 


vi  CONTENTS 

PAGE 

Interest 73 

Ordinary  Interest     ...         .o         ...       73 

Accurate  Interest .77 

To  Find  Time 77 

To  Find  Rate 78 

To  Find  Principal 79 

General  Problems  in  Interest  .         .         .         .         .         .81 

Partial  Payments  . 83 

Bank  Discount 86 

Compound  Interest  .......       89 

Periodic  Interest      ........       90 

Averaging  Accounts 92 

Taxes 99 

Customs  and  Duties         .         .         .         .         .         .         .100 

Insurance 102 

Life  Insurance  .         .         .         .         .         ,         .         .103 

Exchange 105 

Domestic  Exchange  .         .         .         .         .         .         .105 

To  Find  the  Value  of  a  Sight  Draft  .         .         .         .105 

To  Find  the  Value  of  a  Time  Draft  .         .         .         .105 

To  Find  the  Face  of  a  Draft 106 

Foreign  Exchange    .         .         .         .         .         .         ,         .107 

Stocks  and  Bonds 109 

PART   TWO 

Definitions «         .         o         o         1 

Notation 2 

Common  Processes 5 

Addition  —  Integers         .......         5 

Subtraction  —  Integers    .......         7 

Multiplication  —  Integers         ......         9 

Division  —  Integers  .         ,         .         .         .         .         .11 


CONTENTS 


Vll 


PAGE 

Decimals 13 

Addition  —  Decimals        .  .  .  .  .  .  .13 

Subtraction  —  Decimals  .  .  .  .  .  .  .13 

Multiplication  —  Decimals  .  .  .  .  .  .       14 

Division  —  Decimals        .  .  .  .  ,  .  .       1 .5 

Factoring          .         .         .  .  .  .  .  .  .17 

Least  Common  Multiple .  .  .  .  .  .  .18 

Greatest  Common  Divisor  .  .  .  .  .  .19 

Fractions 22 

Changing  to  Lower  Terms         ......       24 

Changing  to  Higher  Terms       .         .         .         .         .         .26 

Changing  an  Improper  Fraction  to  a  Mixed  Number  .       27 

Changing  a  Mixed  Number  to  an  Improper  Fraction       .       28 
Changing  a  Decimal  Fraction  to  a  Common  Fraction       .       29 
Changing  a  Common  Fraction  to  a  Decimal  Fraction       .       30 
Addition  of  Fractions        .  .  .  .         .         .         .31 

Subtraction  of  Fractions  .......       33 

Multiplication  of  Fractions       ......       34 

Division  of  Fractions        .......       37 

Denominate  Numbers 40 

Reduction  of  Denominate  Numbers          ....  40 

Addition  of  Compound  Numbers      .....  43 

Subtraction  of  Compound  Numbers          ....  43 

Multiplication  of  Compound  Numbers     ....  44 

Division  of  Compound  Numbers       .....  44 

Computing  Time     . .  .         .         .         .         .         .         .45 

Aliquot  Parts  .........  46 

Percentage 49 

Profits  and  Losses 57 

Discount        ..........       63 

Trade  Discount         ........       63 

Commission  and  Brokerage 70 


viii  CONTENTS 

PAGE 

Interest         ....        ..o        ...       76 

Accurate  Interest     .         .         ...         .         .         .77 

Ordinary  Interest     ........       79 

.  Explanation      .........       79 

Sixty-Day  Method  —  Ordinary  Interest  Rule  ...       80 
Sixty-Day  Method  —  Accurate  Interest  ....       83 

Commercial  Papers 94 

Partial  Payments 96 

The  United  States  Rule  for  Partial  Payments           .         .       96 
Merchants'  Rule 98 

Bank  Discount 100 

Compound  Interest 104 

Periodic  or  Annual  Interest      .         .         .         .         .         .105 

Averaging  Accounts    .         .         .         .         .         .         .         .108 

General  Principles  of  Average  .         .         .         .         .         .108 

Taxes 116 

Duties  and  Customs  .         .         .         .         .         .         .117 

Insurance      ..........     120 

Life  Insurance  .         .         .         .         .         .         .         .122 

Exchange       ..........     124 

Domestic  Exchange  .         .         .         .         .         .         .124 

Foreign  Exchange .125 

Stocks  and  Bonds 127 

Tables 130 

Index 147 


,      ,  5       J      3  3 

1      3'       3       3        V>        3 

1     3     -,         -,  ,    , 


'       •■    '^     3'    »       '    *•>       '3        '  ' 


RATIONAL  ARITHMETIC 


PART  ONE 

The  following  problems  are  intended  to  afford 
sufficient  practice  to  develop  a  thorough  working 
knowledge  of  practical  business  arithmetic. 

An  effort  has  been  made  to  confine  the  problems  as 
far  as  possible  to  actual  business  conditions  and  to 
present  only  problems  similar  to  those  met  in  actual 
business  experience. 

References  are  by  paragraphs  to  Part  Two  and  are 
sufficiently  copious  to  allow  ready  solution  by  the 
pupil  of  all  problems. 

PRELIMINARY  PROBLEMS 

ADDITION 

The  following  exercises  in  addition  afford  opportunity 
for  frequent  drills.  The  pupil  should  practice  upon 
them  and  similar  problems  provided  by  the  teacher 
throughout  the  course,  or  until  he  is  able  to  add  in 
the  time  specified,  or  in  less  time. 

Study  carefully  paragraphs  96  to  101  inclusive. 

1 


^'•/RATIONAL 

ARITHMETIC 

Practice'  -tintil 

you  ; 

are  able  to 

add  each 

of 

the 

following 

in  15  seconds  or 

•  less : 

1 

2 

3 

4 

5 

6 

324 

196 

596 

287 

812 

285 

436 

289 

321 

422 

263 

635 

243 

781 

284 

389 

426 

149 

429 

423 

675 

674 

529 

728 

182 

317 

329 

263 

198 

463 

327 

262 

918 

721 

984 

824 

148 

425 

786 

416 

623 

296 

283 

348 

465 

129 

467 

179 

2.    Practice  until  you  are  able  to  add  each  of  the 
following  in  25  seconds  or  less : 


7 

8 

9 

10 

11 

12 

324 

463 

247 

472 

289 

521 

642 

721 

962 

749 

394 

347 

^85 

567 

721 

638 

672 

625 

763 

289 

463 

236 

416 

262 

297 

143 

265 

429 

781 

729 

425 

264 

789 

642 

186 

453 

642 

721 

496 

187 

237 

642 

193 

168 

721 

346 

421 

287 

721 

459 

453 

721 

563 

472 

438 

672 

624 

254 

464 

296 

267 

284 

289 

689 

789 

563 

193 

596 

198 

746 

462 

189 

3.    Practice  until  you  are  able  to  add  each  of  the 
following  :n  45  seconds  or  less : 


RATIONAL  ARITHMETIC 


3 


13 

14 

15 

16 

17 

2864 

3829 

4962 

8426 

5479 

4233 

6471 

2794 

7195 

6294 

7185 

1687 

6171 

3824 

1781 

1679 

2762 

2437 

6271 

3326 

2763 

4463 

8263 

2617 

7182 

4638 

9174 

2819 

5409 

4963 

9162 

2896 

6279 

6276 

7126 

2746 

4789 

2854 

2830 

4413 

8427 

6217 

1962 

3418 

7824 

1679 

2834 

7148 

9016 

1671 

2634 

9753 

6523 

2468 

5607 

4791 

2891 

3764 

1695 

2896 

4.    Practice  until  vou  are  able 
following  in  60  seconds  or  less  : 


to  add  each  of  the 


18 

264118 
428307 
711695 
386472 
369143 
642785 
617192 
548237 
167589 
294462 
162781 
146229 
382716 


19 

528563 
742896 
478132 
264389 
1462^27 
584296 
817529 
428127 
362419 
780962 
278438 
261971 
446236 


20 

428137 
298461 
541672 
832744 
167182 
322907 
541891 
851693 
395816 
724594 
280790 
642031 
451682 


21 

252763 
376329 
167251 
146327 
421791 
573619 
287513 
324671 
271293 
348162 
912872 
268047 
634918 


22 

427183 
284562 
711456 
378275 
462871 
146265 
551681 
287354 
851762 
718319 
440892 
632757 
642819 


4  RATIONAL  ARITHMETIC 

5.    Practice  until  you  are  able  to  add  each  of  the 
following  in  75  seconds  or  less : 


23 

24 

25 

26 

27 

416342 

487902 

284062 

614385 

812716 

913457 

226439 

713345 

422716 

341675 

296731 

914362 

167182 

312814 

472386 

284562 

167943 

421671 

567583 

611743 

811706 

208209 

284371 

642217 

296342 

273468 

613317 

176327 

551638 

542138 

296329 

672438 

420416 

281954 

617516 

284672 

719243 

798296 

371621 

271642 

542983 

264738 

146329 

560932 

182133 

718296 

194513 

817043 

174837 

162904 

287981 

382761 

241671 

165329 

816238 

273468 

280642 

146329 

241671 

271609 

28 

29 

30 

31 

32 

24627 

93281 

27682 

14632 

91387 

48231 

44638 

38225 

71136 

26785 

62783 

71289 

16781 

28483 

54321 

24167 

45642 

91483 

43729 

28654 

85262 

71483 

82675 

52847 

43832 

71843 

19721 

46721 

37625 

71819 

29636 

54163 

53482 

54783 

44623 

71083 

27386 

27624 

29654 

48729 

27642 

16721 

14729 

18729 

17453 

29827 

28294 

61453 

47387 

37529 

16429 

26783 

27185 

26475 

62745 

54540 

54296 

16291 

83267 

71258 

68296 

78287 

54385 

29453 

54183 

RATIONAL  ARITHMETIC 


33 

34 

35 

36 

$6743.76 

$3462.78 

$4527.82 

$7345.60 

2846.75 

5287.95 

3675.18 

2847.29 

8421.62 

6379.86 

2643.89 

5640.36 

7329.44 

7429.80 

1796.97 

7281.28 

3780.50 

5463.29 

4238.54 

1267.43 

2894.62 

7128.42 

2879.36 

3629.75 

7481.13 

3864.19 

7481.29 

4678.37 

2563.27 

7133.64 

3279.81 

8126.42 

6247.16 

4461.72 

4782.63 

3726.42 

3729.42 

9562.70 

2871.54 

4671.38 

8427.16 

4683.49 

7627.18 

6275.29 

2945.71 

7216.30 

2375.62 

5463.72 

4453.75 

3200.50 

4671.16 

2963.47 

37 

38 

39 

40 

$3678.44 

$5617,81 

$2896.75 

$8297.50 

2917.53 

2976.37 

4429.36 

6384.96 

6279.45 

3728.44 

1785.29 

7185.16 

1468.71 

1671.38 

6271.38 

4183.94 

2783.94 

5418.75 

1862.45 

5467.28 

7162.85 

2763.48 

5483.92 

4392.40 

1671.36 

6378.27 

2861.57 

5671.39 

2768.29 

5423.75 

4675.38 

2716.42 

5419.75 

1627.30 

9863.75 

3895.16 

1683.27 

4862.19 

5728.50 

5482.95 

4462.91 

7143.84 

6275.81 

7436.81 

2789.65 

3627.52 

3829.62 

2918.15 

3601.82 

5462.79 

7426.85 

4183.95 

4618.79 

2183.96 

5387.09 

6343.19 

4844.60 

4528.17 

6425.70 

2874.67 

6  RATIONAL  ARITHMETIC 

6.    Practice  until  you  are  able  to  add  each  of  the 
following  in  90  seconds  or  less : 


41 

42 

43 

44 

45 

283 

487 

275 

438 

219 

347 

365 

381 

622 

736 

462 

791 

456 

563 

432 

728 

384 

179 

729 

275 

529 

429 

527 

247 

863 

633 

186 

623 

384 

179 

387 

795 

819 

862 

327 

472 

324 

942 

721 

453 

694 

432 

163 

903 

618 

458 

175 

209 

415 

721 

182 

293 

725 

541 

153 

791 

764 

483 

287 

286 

453 

379 

342 

453 

723 

645 

453 

671 

628 

429 

286 

186 

538 

143 

186 

729 

791 

827 

517 

791 

452 

428 

113 

386 

917 

628 

384 

452 

295 

283 

139 

292 

938 

672 

418 

255 

574 

286 

286 

726 

386 

387 

721 

295 

814 

675 

198 

453 

721 

428 

341 

421 

618 

193 

297 

296 

502 

721 

382 

548 

416 

218 

453 

453 

353 

209 

721 

618 

182 

287 

381 

347 

721 

763 

186 

RATIONAL  ARITHMETIC  7 

7.  Pupils  who  have  acquired  proper  facihty  in  the 
preceding  problems  will  have  no  difficulty  in  per- 
forming the  following  with  sufficient  rapidity  : 


$1283.64 

$  284.16 

$  94.43 

$3728.54 

785.30 

1728.32 

2168.75 

6241.57 

2721.83 

4278.19 

1413.80 

287.63 

1763.20 

2674.19 

287.60 

729.42 

487.32 

905.16 

782.19 

98.17 

9.05 

728.40 

3187.42 

46.54 

87.62 

7.14 

4163.27 

19.72 

44.33 

287.64 

973.29 

83.71 

1286.75 

4239.17 

4871.30 

129.40 

343.06 

2476.28 

2972.43 

987.60 

428.06 

287.19 

187.62 

2763.42 

71.37 

458.16 

98.43 

1486.71 

289.70 

6724.13 

642.19 

4938.27 

2729.40 

8.60 

3894.16 

9132.38 

2876.42 

278.40 

721.32 

764.20 

29.00 

4291.62 

193.64 

381.65 

427.40 

71.85 

4287.16 

28.53 

9.90 

13  63 

1328.72 

9.19 

727.62 

2842.75 

46.84 

2.85 

3478.24 

4871.20 

13.16 

642.38 

9288.75 

389.45 

9.68 

94.73 

2471.05 

642.16 

171.24 

2652.81 

287.63 

71.00 

283.95 

453.17 

274.28 

9  80 

90.60 

287.29 

297.62 

2621.13 

468.13 

4182.19 

48.39 

38.79 

71.24 

78.50 

184.19 

178. '3 

457.62 

1246.53 

8 


RATIONAL  ARITHMETIC 


8.    Complete  the  following  statement  by  finding  the 
totals  of  columns  and  the  totals  from  left  to  right : 


Mar.  1 

2 

3 

4 

5 

7 

8 

9 

10 

11 

13 

14 

15 

16 

17 

18 

20 

21 

22 

23 

24 

25 

27 

28 

29 

30 

31 

Totals 


Corn 


284.65 
78.39 
164.70 
348.62 
175.29 
98.40 
467.28 
298.70 
587.64 
298.63 
728.45 
687.44 
285.90 
429.18 
697.65 
597.67 
738.42 
914.16 
576.80 
894.48 
1098.27 
384.62 
725.30 
456.82 
689.63 
521.16 
782.73 


Wheat 


487.62 
629.34 
587.19 
984.63 
729.40 

1180.68 
960.27 
450.18 
829.30 
486.90 
752.83 
647.92 
473.85 
738.24 
287.16 
587.90 
699.48 
826.16 
487.28 
745.60 
952.75 
397.26 
678.40 

1238.50 
927.24 
627.42 
795.38 


Oats 


125.30 
326.45 
217.16 

98.43 
454.20 

90.12 

72.15 
248.00 
316.90 
109.09 
402.14 

48.30 
178.85 
148.16 
242.90 
371.42 
416.78 
212.13 

98.42 
248.16 
500.00 
187.19 
238.16 

71.45 
122.58 
108.12 
314.60 


Apples 


285.64 
171.80 
219.17 
287.60 
58.09 
171.23 
264.12 
175.11 
147.16 
209.04 
348.17 
250.60 
658.12 
347.60 
795.80 
721.80 
424.70 
368.40 
519.67 
295.48 
487.29 
368.12 
563.27 
448.12 
219.60 
348.75 
218.90 


Beans 


697.60 
721.80 
450.67 
385.90 
287.40 
458.62 
721.30 
318.90 
427.65 
386.37 
516.80 
428.54 
719.80 
579.60 
387.40 
473.20 
517.68 
429.90 
343.80 
624.39 
387.26 
419.72 
611.41 
516.75 
218.19 
365.40 
409.08 


Flour 


48.60 

75.60 

308.00 

9.50 

24.72 
317.80 
240.16 
421.75 

48.90 

97.50 
105.12 
215.70 

84.95 
198.60 
385.00 

95.16 
171.20 
384.60 
142.80 
218.70 
140.68 
214.09 
190.54 
138.20 

97.42 
164.83 
120.95 


Totals 


SUBTRACTION 


While  no  practice  seems  necessary  on  simple  sub- 
traction of  integers,  the  pupil  should  read  carefully 
102  to  109  inclusive,   and  then  balance  the  following 


RATIONAL  ARITHMETIC 


9 


accounts  according  to  the  method   explained    in    108 
and  109. 


9.    Balance  the  following  accounts  : 
1.    Dr.  QUAKER  OATS   CO. 


Cr. 


<2A--e-^<- 


$76.50 
16.75 
39.75 


2.    Dr. 


D.   T.   AMES   &   CO. 


Cr. 


$185.25 

$603.75 

8. 

215.50 

2.10 

73.94 

121.50 

-   24. 

3.    Dr. 


INTEREST 


Cr. 


$1. 

$  .68 

7.70 

9.21 

2.52 

10. 

3.70 

1.40 

1.12 

10 


RATIONAL  ARITHMETIC 


4.    Dr. 


F.   B.    SMITH 


Cr, 


$  6.75 

$33.45 

3.75 

6. 

14.50 

10.25 

12.25 

5.40 

13.25 

5.    Dr. 


SALES 


Cr. 


$12.20 

$   6.60 

4.65 

1143.75 

5.10 

843.19 

4.85 

6. 

8. 

5.50 

6.    Dr. 


PARK  &  STEWART 


Cr. 


$1021.65 

$589.05 

964.41 

56.40 

558.72 

236.25 

280. 

661.50 

RATIONAL  ARITHMETIC 


11 


7.    Dr. 


JAMES   CARTER 


Cr. 


$  1.91 

$278.60 

716. 

956.20 

733.13 

8.20 

588.03 

9.16 

436. 

833.70 

47.31 

8.    Dr. 


BLANCHARD   &   CO, 


Cr. 


$  16.50 

$  3.60 

7.64 

10.80 

18.50 

7.75 

219.21 

19.60 

12.50 

4.23 

9.    Dr. 


F.   G.  MILLER 


Ci\. 


$63.60 

$  4.17 

25. 

.60 

14.50 

140. 

25. 

207.50 

21.15 

55. 

16.67 

9. 

61.50 

12 

10.    Dr. 


RATIONAL   ARITHMETIC 


CHARLES   SMITH 


CV. 


$  76. 

$  51.01 

98.55 

60.03 

83.02 

171. 

59.20 

80.61 

105.08 

2.10 

43.20 

210.33 

13.10 

29.76 

Dr. 


H.   A.   THOMAS 


Cr. 


$1240.50 

$  500. 

876. 

750. 

453.35 

1250. 

96.73 

2575. 

1000. 

354.56 

Dr. 


SAWLER  &   HASKINS 


Cr. 


$1246.34 

878.14 

2543.65 

3746.32 


$2354.56 
3143.42 
1245.15 
1873.94 


DECIMALS 


MULTIPLICATION 


The  object  of  these  exercises  is  to  acquire  accuracy 
in  locating  the  decimal  line. 


References  133,  134,  135. 

10.      1.  .974X.35  = 

2.  8.2X9.6  = 

3.  284X.75  = 

4.  346.5X10.02  = 

5.  27X.38  = 

6.  .1655X18  = 

7.  .4355X16.66  = 

8.  844.5X7.404  = 

9.  1263X8.04  = 

10.  .44X8.05  = 

11.  65.410X.585  = 

12.  75000X.0098  = 

13.  5.9X26.7362  = 


14.  75X6.0053  = 

15.  3.926X464  = 

16.  4.872X.386  = 

17.  84672X8.4  = 

18.  2.8973X. 80806 

19.  .94X100.82  = 

20.  446.8X3.044  = 

21.  287X9.0104  = 

22.  .9634X58  = 

23.  283.86X.396  = 

24.  94.652X4.87  = 

25.  84X. 000238  = 


DIVISION 


Study  136  to  141  inclusive. 

In  solving  the  following  problems  make  strict  applica- 
tion of  the  rules  given  in  137  and  138. 

13 


14 


RATIONAL   ARITHMETIC 


Paragraphs   139,    140, 
method  of  sokition. 

11.    1.  14.875^3.5  = 

2.  338.52^8.4  = 

3.  1385.128-^21.6  = 

4.  3.456^12  = 

5.  654.5  H- 11  = 

6.  2.464-7-1100  = 

7.  43.4172^74.6  = 

8.  .01581 -^  .255  = 

9.  6.305 -=-3.25  = 

10.  8.63-^3.84  = 

11.  79.896  H- .53264  = 

12.  372.012-^58  = 

13.  14.157-^2.6  = 

14.  14.157-^.26  = 

15.  1284.7^.3875  = 

16.  246.9^23  = 

17.  1640.625^1875: 

18.  286.996 -^  .914  = 

19.  17.408-^12.8  = 

20.  6264-^.348  = 

21.  14.825^-8.29  = 

22.  286.327 -^  156  = 

23.  16.38^.284  = 

24.  284.62^84  = 

25.  29.728-^8.4  = 


and   141   fully   illustrate   the 

26.  1.75 -^  23.5765  = 

27.  437.8675^23.8  = 
=     28.  23.183^19.7463  = 

29.  246.-^7.875  = 

30.  75-^125  = 

31.  1.8675^5.75  = 

32.  124.56^15.8  = 

33.  48567.75^4.875  = 

34.  76.50^1250  = 

35.  .5863^12.52  = 

36.  .9875 -^. 584  = 

37.  23.45675 -^  1.375  = 

38.  23.-^27  = 

39.  125.^56  = 

40.  324. -T- 678  = 

41.  12.875-^4.25  = 

42.  125.6^2^7.75  = 

43.  415.875^.1275  = 

44.  234. 15 -r- 5.875  = 

45.  56.625^-128.50  = 

46.  153.8756^53.962  = 

47.  346.4278^8.4695  = 

48.  16.4584-^3.4565  = 

49.  125.4632-7-18.4965  = 

50.  4356.4589-^27.4875  = 


FRACTIONS 


Study  carefully  IGO  to  180  inclusive. 
References  181,  18^2,  183. 

12.    Reduce  the  following  to  lowest  terms : 


1. 

7  20 
12  00 

6. 

2. 

6  30 
2  8  35 

7. 

3. 

9  1  0 

104  0 

8. 

4. 

105 
2  3  1 

9. 

5. 

2  1  0 
A  5  5 

10. 

35 
45 

16  08 

_6JL5_ 
12  00 

LJ.5. 

'7  8  2 

2  3  2 


11. 


12. 


13. 


5  5  1 


u 

9 
12 

Q2 

8 


71 

14.  ^ 
25 

15.  ^ 
18 


References  184,  185,  1 80. 

13.  Change  the  following  fractions  to  the  denomina- 
tions designated : 

1.  ttol25ths.  6.  tto88ths. 

2.  I  to  56ths.  7.  tV  to  1524ths. 

3.  i  to  72ds.  8.  f  to  126ths. 

4.  A  to  195ths.  9.  A  to  352ds. 

5.  f  to  84ths.  10.  i  to  27ths. 

References  187,  188,  189. 

14.  Change  the  following  to  mixed  numbers  : 


1  2J7 


3. 


1-6 
9 


2. 


1  23 
5 


A.         124 


5. 

6. 
15 


5  1  5 

16 

3  2  3 
1  5 


7. 

8. 


4  27 
9 

]  2  5 


16  RATIONAL   ARITHMETIC 

References  190,  191,  192. 

15.  Change  the  following  to  improper  fractions  : 

1.  If  4.    9t  7.    81i  10.    214f 

2.  4i  5.    23f  8.    371  11.    34 A 

3.  8|  6.    46|  9.    123*  12.    43| 

References  193,  194,  195. 

16.  Change  the  following  to  common  fractions  or 
mixed  numbers : 

1.  .375  '       5.  .15625  9.  .66f 

2.  .875  6.  8.25  10.  .272t\ 

3.  .0625  7.  17.875  11.  1.77J 

4.  .125  8.  9.28125  12.  .384t'^ 

References  196,  197,  198,  199,  200. 

17.  Change  the  following  to  decimal  equivalents : 


1. 

3 
4 

5. 

4 
9 

9. 

1 1 

17 

13. 

8i 

2. 

4 
5- 

6. 

5 
6^ 

10. 

9 
2^3" 

14. 

9i3 

3. 

5 

8 

7. 

7 
8 

11. 

3f 

16. 

12| 

4. 

1  1 
1  3 

8. 

9 
1  3 

12. 

4f 

ADDITION   OF   FRACTIONS 
References  201,  202,  203,  204,  205. 


18. 


1. 

3  1     5  _ 

4  1     8  "" 

2. 

7  14          1   _ 

8  1     5          2  — 

3. 

1              3              5     _ 
13          2  4          1  8  ~~ 

4. 

12S+8f+27|  = 

RATIONAL   ARITHMETIC  17 

6.  21t+46M+29t  = 

7.  628A+56f+16i+30A  = 

8.  34i+26f +91x^0 +  631  = 

9.  4f+27oV+33i+12iJ  = 
10.  7|+9i+15A+5i  = 

SUBTRACTION   OF  FRACTIONS 

References  206,  207. 

19.    1.   1-1=  5.    146^-791  = 

2.  12f-7|=  6.    1246^^-9831  = 

3.  214H-185i=  7.    25f-14f  = 


20. 


4.    ^5i-5i=  8.    3461 -217 A 

MULTIPLICATION   OF  FRACTIONS 
References  208  to  218  inclusive,  and  135. 


5     


1. 

4  v  5  — 

11. 

319tXl8f  = 

2. 

12iX8  = 

12. 

2.15iX24  = 

3. 

124X161  = 

13. 

24X.12i  = 

4. 

218x29f  = 

14. 

246X.16f  = 

5. 

289fXl26  = 

15. 

814X.44i  = 

6. 

1^2"X03"  = 

16. 

31.62iX8f  = 

7. 

124iX27f  = 

17. 

312x24.08i  = 

8. 

128fVX23i  = 

18. 

459X3121  = 

9. 

5122^X137  = 

19. 

12461X34.52  = 

10. 

461X43  = 

20. 

4.16fX.12i  = 

18  RATIONAL  ARITHMETIC 

DIVISION    OF   FRACTIONS 
References  219  to  225  inclusive,  and  139,  140,  141. 

21. 


1. 

4-5  

5     •     8  ~ 

11. 

246f^l3f  = 

2. 

1  2     •     8  _ 
1  9    •     9  ~ 

12. 

12461  :  41  = 

3. 

7     •     5  _ 
8*6 

13. 

4181-151  = 

4. 

125^1  = 

14. 

2161-^19  = 

5. 

24  :  1  = 

15. 

1248^  :  27^  = 

6. 

346 --1  = 

16. 

456fV-M25  = 

7. 

1246^151  = 

17. 

5. 141 --23  = 

8. 

482 --171  = 

18. 

4246  :  .171  = 

9. 

8461 -^  26  = 

19. 

128.571  :  .12^  = 

10. 

5321  ^  18 1^  = 

20. 

43.55f^.l6f  = 

PRACTICE  PROBLEMS  INVOLVING  THE  USE  OF  FRACTIONS 

AND  DECIMALS 

The  following  problems  are  intended  to  show  the 
application  of  the  general  principles  of  common  frac- 
tions. Their  proper  solution  involves  a  knowledge  of 
paragraphs  160  to  225  inclusive. 

22.  1.  Four  pieces  of  cloth  measure  respectively 
311  yd.,  43i^  yd.,  5Q^  yd.,  and  44i  yd.  What  is  the 
total  length.^ 

2.  What  is  the  sum  of  23.8t%,  32.35f,  56|,  194,  and 
i?     Carry  to  the  fourth  decimal  place. 

3.  I  am  about  to  ship  a  box  containing  20f  lb. 
coffee,  3^6  lb.  tea,  23^  lb.  ham  and  16f  lb.  bacon  to  my 
camp.  If  the  box  weighs  2f  lb.,  what  is  the  total 
weight  of  the  shipment? 


RATIONAL  ARITHMETIC  19 

4.  A  grocer  bought  six  bags  of  coffee,  weighing 
respectively  132f  lb.,  128^  lb.,  127f  lb.,  136|  lb., 
134f  lb.,  and  128|  lb.  Allowing  H  lb.  for  the  weight 
of  each  bag,  what  would  be  the  total  net  weight  of  the 
coffee  ? 

5.  I  bought  5  barrels  of  sugar.  The  net  weight  of 
each  respectively  was  275t  lb.,  283i  lb.,  2711  lb., 
293f  lb.,  and  2851  lb.     Find  the  total  net  weight. 

6.  I  bought  a  f  interest  in  a  bowling  alley,  and  sold 
my  brother  a  -re  interest.     How^  much  do  I  own  ? 

7.  An  automobilist  on  a  tour  completes  -^e  of  the 
trip  on  the  first  day,  i  on  the  second  day,  and  i  on  the 
third  day.  He  then  finds  himself  350  miles  from  his 
destination.  What  is  the  total  length  of  the  trip  and 
how  far  has  he  alreadv  advanced  ? 

8.  I  bought  a  house  for  $7500.  I  paid  i  of  the  pur- 
chase price  in  cash,  f  of  the  remainder  was  paid  in  six 
months,  and  I  am  now  ready  to  make  the  final  pay- 
ment.    For  what  sum  must  mv  check  be  WTitten  .^ 

9.  I  can  do  a  piece  of  work  in  5  da  vs.  Mv  brother 
requires  7  days  to  do  the  same  thing.  If  we  work 
together  how  long  will  it  take  to  complete  the  job  ? 

10.  If  coffee  loses  yq  of  its  weight  in  roasting,  how 
many  pounds  of  green  coffee  must  be  roasted  to  pro- 
duce 375  lb.  ? 

11.  A  farmer  bought  a  cow  for  $52f  and  a  ton  of  hay 
for  $29f .  How  much  change  would  he  receive  out  of 
Si  one-hundred-dollar  bill  ? 

12.  A  bookkeeper's  pay  envelope  contains  three 
$10's,  one  $5,  one  $2,  and  one  $1.     He  paid  for  board 


20  RATIONAL  ARITHMETIC 

$1H,  a  bill  amounting  to  $8f ,  and  bought  a  hat  for  $3,  a 
pair  of  gloves  for  $lf ,  and  two  pairs  of  stockings  at  $.50 
a  pair.     What  part  of  his  week's  pay  did  he  have  left  ? 

13.  What  will  \l-r2  dozen  eggs  cost  at  $.58f  per 
dozen  ? 

14.  If  71  tons  of  hay  cost  $182^,  what  will  llf  tons 
cost  ? 

15.  I  bought  4375f  bushels  of  corn  at  $.80f  a  bushel, 
and  2350^  bushels  of  oats  at  $.61f  a  bushel.  What 
was  the  entire  investment.'^ 

16.  Find  the  total  cost  of  the  following:  350  lb. 
Rio  coffee  at  $.47| ;  450  lb.  Mocha  coffee  at  $.41f ; 
900  lb.  white  sugar  at  $.10f ;  900  lb.  brown  sugar  at 
$.09f;  970  lb.  granulated  sugar  at  $.08f;  172  lb. 
butter  at  $.56|. 

17.  A  merchant  sold  80  lb.  of  butter  at  $.57f ;  43 
dozen  of  eggs  at  $.61f  per  dozen ;  32^  gallons  of  milk 
at  $.60  a  gallon.     What  was  the  total  amount  of  sales  ? 

18.  A  piece  of  cloth  containing  47f  yd.  was  sold  for 
$9. 94 J.     What  was  the  price  per  yard  ? 

19.  One- third  of  a  firm's  capital  is  invested  in 
merchandise,  three-eighths  in  real  estate,  and  the  rest, 
$18,200,  is  cash.  What  is  the  capital  of  the  firm.^ 
How  much  is  invested  in  merchandise,  and  how  much 
in  real  estate  ? 

20.  A  farm  yields  96.08  bushels  of  potatoes  to  the 
acre,  36.625  bushels  of  oats  per  acre,  15.52  bushels  of 
wheat  per  acre.  156  acres  were  planted  in  potatoes, 
214  acres  in  oats,  and  19.3  acres  in  wheat.  What  is  the 
total  number  of  bushels  harvested  ? 


Rx\TIONAL  ARITHMETIC  21 

21.  I  have  withdrawn  J  of  my  money  from  the  bank 
and  have  $376.40  remaining.  How  much  did  I  with- 
draw? 

22.  A  partnership  consists  of  three  members  who  in- 
vest respectively  i,  i,  i  of  the  capital,  and  agree  to  share 
losses  and  gains  in  the  same  proportion.  How  much  will 
be  each  partner's  share,  if  there  is  a  profit  of  $13,416.75  ? 

23.  A  business  man  finds  himself  unable  to  meet  his 
entire  obligations.  He  owes  $12,360  and  has  $10,300 
available  with  which  to  pay.  What  part  of  his  lia- 
bilities can  he  meet?  How  many  cents  on  the  dollar 
is  this  ? 

24.  A  invested  i  of  the  capital  of  a  firm,  B  i,  C  i, 
and  D  the  remainder.  D's  share  is  $1690.  What 
was  A's,  B's,  and  C's  investment? 

25.  A  house  and  lot  cost  $6600.  The  house  costs 
i  more  than  the  land.     What  was  the  cost  of  each? 

26.  How  manv  bushels  is  .75  of  640  bushels? 

27.  A  merchant  sold  162  barrels  of  flour  which  is  f 
of  his  stock  of  flour.     How  much  flour  had  he  at  first  ? 

28.  A  merchant  sold  480  barrels  of  flour  which  is  .625 
of  his  entire  stock.     Hov/  many  barrels  had  he  at  first  ? 

29.  A  man,  at  his  death,  left  $30,000  to  his  wife,  son, 
and  daughter ;  .5  of  this  sum  went  to  his  wife,  .375 
to  his  daughter,  and  .125  to  his  son.  How  much  did 
each  receive  ? 

30.  I  have  just  learned  that  one  of  my  customers  has 
failed  and  is  able  to  pay  only  $.525  on  the  dollar.  My 
claim  against  him  amounted  to  $134.40.  How  much 
will  I  receive? 


DENOMINATE   NUMBERS 

Reference  ^29. 

23.  1.    Reduce  £34,  8^,  7d  to  pence. 

2.  Reduce  4  T.,  5  cwt.,  85  lb.  to  pounds. 

3.  Reduce  14  gal.,  3  qt.,  1  pt.  to  pints. 

4.  Reduce  1  cwt.,  24  lb.,  3  oz.  to  ounces. 

5.  Reduce  1  da.,  3  hr.,  25  min.  to  seconds. 

6.  Reduce  14  yr.,  5  mo.,  3  wk.  to  days. 

7.  Reduce  1  m.  25  rd.,  4  yd.,  2^  ft.  to  inches. 

8.  Reduce  2  hhd.,  14  gal.,  3  qt.  to  pints. 

9.  Reduce  14  bu.,  3  pk.  to  pints. 

10.    Reduce  3  A.,  2  sq.  rd.,  10  sq.  yd.  to  sq.  ft. 

Reference  230. 

24.  1.    Reduce  3462  sq,  in.  to  higher  denominations. 

2.  Reduce  14>6Sd  to  higher  denominations. 

3.  Reduce  17696  lb.  to  higher  denominations. 

4.  Reduce  32625  gr.  to  higher  denominations. 

5.  Reduce  12760  in.  to  higher  denominations. 

6.  Reduce  18428  sq.  in.  to  higher  denominations. 

7.  Reduce  3896  cu.  ft.  to  higher  denominations. 

8.  Reduce  4843c?  to  higher  denominations. 

9.  Reduce  120615  sec.  to  higher  denominations. 
10.  Reduce  633  pt.  to  higher  denominations. 

22 


RATIONAL  ARITHMETIC  23 

Reference  231. 

25.  1.    Reduce  .327  m.  to  lower  denominations. 

2.  Reduce  .35  hr.  to  lower  denominations. 

3.  Reduce  .875  yd.  to  lower  denominations. 

4.  Reduce  .135  yr.  to  lower  denominations. 

5.  Reduce  f  mo.  to  lower  denominations. 

6.  Reduce  .125  m.  to  lower  denominations. 

7.  Reduce  £2.3456  to  lower  denominations. 

8.  Reduce  £12.456  to  lower  denominations. 

9.  Reduce  f  m.  to  lower  denominations. 
10.  Reduce  tt  yr.  to  lower  denominations. 

Reference  232. 

26.  1.    Reduce  4  yd.,  2  ft.  to  a  decimal  of  a  rod. 

2.  Reduce  3  gal.,  2  qt.,  1  pt.  to  gallons. 

3.  Reduce  2  pk.,  3  qt.,  1  pt.  to  a  decimal  of  a 

bushel. 

4.  Reduce  35  min.,  18  sec.  to  a  decimal  of  a  day. 

5.  Reduce  18  rd.,  4  yd.,  2  ft.  to  rods. 

6.  Reduce  4  cwt.,  85  lb.  to  a  decimal  of  a  ton. 

7.  Reduce  Ss,  lOd,  2/  to  a  decimal  of  a  pound. 

8.  Reduce  18  sq.  rd.,  4  sq.  yd.  to  a  decimal  of 

an  acre. 

9.  Reduce  14  hr.,  35  min.,  10  sec.  to  a  decimal 

of  a  day. 
10.    Reduce  185  lb.,  12  oz.  to  a  decimal  of  a  ton. 


ALIQUOT  PARTS 

Study  paragraphs  240  and  241,  memorizing  the  table 
and  noting  appHcation  as  explained  in  note  a. 

Reference  241. 
27.    Find  the  cost  of  : 

1.  720  1b.  at  50^;   at  33J^ ;   at  25^. 

2.  120  lb.  at  33i^;   at  25^;   at  20^;   at  12i^. 

3.  360  lb.  at  6U  \   at  6f  ^  ;   at  10^  ;   at  12i^. 

4.  840   yd.    at    10^;     at    12^^;     at    14f  ^ ;     at 

25^. 

5.  4800  lb.  at  8^^ ;  at  6i^;  at  12^^;    at  IGf^; 

at  10^. 

6.  240  yd.  at  8i^  ;   at  6f  ^  ;   at  10^  ;   at  12^^. 

7.  2480  yd.  at  25^;    at  50^;    at  33i^ ;    at  20^. 

8.  480  yd.  at  6i^  ;    at  8^^  ;    at  6f  ^  ;    at  10^  ; 

at  12i^. 

9.  560  yd.  at  8J^  ;    at  6i^  ;    at  6f  ^  ;    at  10^; 

at  12i^. 

10.  204    yd.    at   50^;     at   33^^;     at   25^;     at 

11.  4200  yd.   at   10^;    at    12^^;    at   14f  ^ ;    at 

16f^;   at  25^. 

24 


RATIONAL   ARITHMETIC  25 

12.  1800  lb.  at  nW.   at  16f^;   at  20^;   at  25^; 

at  33i^. 

13.  1500  yd.  at  $1 ;   at  12J^  ;   at  14f  ^  ;   at  16f^; 

at  25^. 

14.  490  doz.  at  12i^ ;   atlOf^;    at  20^;   at  6f  ^. 

15.  960  yd.  at  8i^ ;    at  6i^;    at  10^;  at  UW, 

at  6f  ^. 

Reference  241. 

28.    Find  the  total  cost  of  : 

1.    38  lb.  at  25^  2.    63  yd.  at  28^^ 

84  lb.  at  37ijzi  81  yd.  at  33i^ 

72  lb.  at  75^  18f  gr.  at  52^ 

48  lb.  at  41|^  28  doz.  at  50^ 

96  lb.  at  33i^  58  yd.  at  14f  ^ 

24  lb.  at  12i^  235  yd.  at  40^ 

3.  61  lb.  at  48^  4.  25  bu.  at  96<^ 
480  lb.  at  16f  ^       20  bu.  at  88jZ^ 

25  lb.  at  44^         31i  bu.  at  $2 
161  lb.  at  48^        50  bu.  at  $11.50 
240  lb.  at  18f  ^        12i  bu.  at  $3.60 
72  lb.  at  37i^        25  bu.  at  $1.64 

5.  25  yd.  at  76^^  6.  37i  bu.  at  72^ 

37^  yd.  at  96^  75  bu.  at  $3.20 

750  yd.  at  12^^  62^  bu.  at  $1.36 

168  doz.  at  12i^  14f  bu.  at  $1.54 

420  yd.  at  33i^  50  bu.  at  $5.85 

176  yd.  at  31i^  12i  bu.  at  $.64 


26 


RATIONAL  ARITHMETIC 


Reference  241. 
29.    Find  the  cost  of  : 

1.  6i  A.  land  at  $192. 

2.  125  lb.  tea  at  48^. 

3.  34  lb.  tea  at  50^. 

4.  25  lb.  coffee  at  44^. 

5.  25  T.  coal  at  $10.80. 

6.  72  pieces  lace  at  $1.25. 

7.  44  yd.  velvet  at  $2.50. 

8.  2i  bu.  potatoes  at  $1.48. 


9.  12i  bu.  turnips  at  74^. 

10.  12i  yd.  silk  at  $1.04. 

11.  84  tables  at  $12.50. 

12.  36  sets  chairs  at  $125, 

13.  12i  yd.  linen  at  56^. 

14.  25  pieces  lace  at  $6.60. 

15.  62i  T.  coal  at  $9.50. 

16.  375  T.  coal  at  $11.50. 


17.  264  A.  land  at  $37.50. 

18.  320  bu.  potatoes  at  $2.12^. 

19.  810  T.  coal  at  $12.50. 

20.  1250  bbl.  pork  at  $24. 

21.  1280  lb.  rice  at  12^^. 

22.  366  yd.  silk  at  $1.66f. 

23.  Hi  yd.  duck  at  36^. 

24.  474  gal.  cider  at  33i^. 

25.  1680  qt.  vinegar  at  16f  ^. 

26.  648  lb.  sugar  at  10^. 

27.  208  yd.  tape  at  2i^. 

28.  176  lb.  tea  at  50^. 

29.  1742  yd.  silk  at  $1.50. 

30.  560  gal.  oil  at  12i^. 

Reference  241. 

30.   Find  the  total  cost  of  the  following : 


RATIONAL   ARITHMETIC  27 

1.  180  \h.  at  SSU  2.  138  lb.  at  33i^ 

760  lb.  at  25^  7^28  lb.  at  62^^ 

54  lb.  at  37i^  224  lb.  at  25^ 

144  lb.  at  33i^  960  lb.  at  66f  ^ 

72  lb.  at  37i^  72  lb.  at  12^^ 

150  lb.  at  66f  f!^  904  lb.  at  87^^ 

3.  196  yd.  at  16f  ^  4.  147  yd.  at  55|^ 

180  yd.  at  66|^  24  yd.  at  66f  ^ 

288  yd.  at  33i^  28  yd.  at  75^ 

459  yd.  at  IH^  84  yd.  at  25^ 

72  yd.  at  25^  56  yd.  at  12^^ 

48  yd.  at  16S^  48  yd.  at  75^ 

183  yd.  at  33^^  246  yd.  at  25^ 

5.  66  gal.  at  33i^  6.  441  gal.  at  55U 

64  gal.  at  87i^  3248  gal.  at  6U 

63  gal.  at  llU  H^  gal-  at  22f  ^ 

144  gal.  at  83^^  266  gal.  at  28^^ 

91  gal.  at  71f  ^  384  gal.  at  62^^ 

96  gal.  at  62^^  368  gal.  at  31i6 

16  gal.  at  87i^  248  gal.  at  87^^ 

945  gal.  at  55U  ^^^  gal.  at  83ij^ 

7.  750  yd.  at  33^^  8.  648  yd.  at  QU 

427  yd.  at  42f  ^  684  yd.  at  33i^ 

87i  yd.  at  50^  496  yd.  at  75^ 

52  yd.  at  62^^  186  yd.  at  83i^ 

450  yd.  at  6f  ^  125  yd.  at  18^ 

2112  yd.  at  SU  144  yd.  at  37i^ 

240  yd.  at  SU  297  yd.  at  444^ 

174  yd.  at  16f  ^  287  yd.  at  UU 

249  yd.  at  25^  918  yd.  at  33i^ 


28  RATIONAL  ARITHMETIC 

9.  144  lb.  at  83i^    10.  144  lb.  at  16f  ^ 

480  lb.  at  ^lU  216  lb.  at  m^ 

282  lb.  at  83i^  872  lb.  at  12i^ 

312  lb.  at  33i^  348  lb.  at  25^ 

427  lb.  at  71f  ^  72  lb.  at  SU 

184  lb.  at  12i^  186  lb.  at  87^^ 

940  lb.  at  IH  138  lb.  at  334^ 

462  lb.  at  66f  ^  96  lb.  at  QU 

342  lb.  at  16f  ^  384  lb.  at  50^ 

11.  84  yd.  at  91f^     12.  84  lb.  at  58^^ 

288*^  yd.  at  12^^  960  lb.  at  16f  ^ 

345  yd.  at  66|^  728  lb.  at  62i^ 

192  yd.  at  37^^  36  lb.  at  43f  ^ 

423  yd.  at  33i^  64  lb.  at  41f  ^ 

280  yd.  at  12i^  96  lb.  at  12^^ 

324  yd.  at  411^  72  lb.  at  41f  ^ 

284  yd.  at  25^  348  lb.  at  75^ 

396  yd.  at  33^^  246  lb.  at  33i^ 

64  yd.  at  5Q\i  344  lb.  at  37i^ 

13.  87i  yd.  at  $2.48    14.  176  yd.  at  $1.12i 

192  yd.  at  87^^  75  yd.  at  16^ 

28  yd.  at  75^  27  yd.  at  75^ 

Hi  yd.  at  18^  5Q  yd.  at  83^^ 

144  yd.  at  lli^  17  yd.  at  25^ 

25  yd.  at  44^  12^  yd.  at  39^ 

75  yd.  at  24^  72  yd.  at  41|^ 

87i  yd.  at  $2.88  344  yd.  at  37^^ 

270  yd,  at  \\U  ^4  yd.  at  8^^ 

24  vd.  at  75^  87i  yd.  at  88^ 


RATIONAL  ARITHMETIC  29 

15.  11511  lb.  at  20^  16.  156  lb.  at  66f^ 

960  lb.  at  16f  ^  284  lb.  at  25^ 

728  lb.  at  62i^  396  lb.  at  SSU 

32  lb.  at  43f  ^  64  lb.  at  5^^ 

64  lb.  at  ^U  384  lb.  at  SlU 

96  lb.  at  12i^  84  lb.  at  58^^ 

72  lb.  at  41f  ^-  960  lb.  at  16f  ^ 

348  lb.  at  75^  728  lb.  at  62^^ 

246  lb.  at  33i^  96  lb.  at  43|^ 

344  lb.  at  37i^  98  lb.  at  37^^ 

17.  96  yd.  at  SU  18.  594  lb.  at  66f^ 

96  yd.  at  12^^  963  lb.  at  83^^ 

348  yd.  at  75^  312  lb.  at  37i^ 

72  yd.  at  41f  ^  251  lb.  at  50^ 

246  yd.  at  33^^  603  lb.  at  11^^ 

344  yd.  at  37^^  552  lb.  at  66f  ^ 

156  yd.  at  66f  ^  133  lb.  at  14f  ^ 

132  yd.  at  91f  ^  528  lb.  at  61^ 

84  yd.  at  50^  273  lb.  at  83^^ 

328  yd.  at  25^  368  lb.  at  31i^ 

• 

19.  146  gal.  at  37^^  20.  200  yd.  at  37i^ 

245  gal.  at  42f  ^  384  yd.  at  18f  ^ 

672  gal.  at  16f  ^  288  yd.  at  83i^ 

18  gal.  at  87i^  294  yd.  at  mU 


162  gal.  at  16f  ^  918  yd.  at  44^^ 

332  gal.  at  18 J^  459  yd.  at  11^^ 

369  gal.  at  33i^  18  yd.  at  12i^ 

828  gal.  at  311^  111  yd.  at  33i^ 

693  gal.  at  16f  ^  164  yd.  at  62i^ 

918  gal.  at  44|c^  8  yd.  at  87^^ 


30 


RATIONAL  ARITHMETIC 


EXERCISES   IN   BILLING 
Reference  241. 

31.    1.    Copy  and  extend  the  following  bill : 

Salem,  Mass.,  July  20,  1919 
Mr.  J.  A.  Brown, 

3  Leach  St., 

City. 

To  S.  S.  Pierce  &  Co.,  Dr. 

Terms  Cash 


8  bu.  beans 

@  $3.75 

108  lb.  butter 

@      .50 

84  lb.  cheese 

@      .33i 

129  doz.  eggs 

@      .50 

150  lb.  lard 

@      .42f 

25  bu.  potatoes 

@   2.16 

72  lb.  rice 

@      .161 

12  lb.  Japan  tea 

@      .5Q\ 

360  lb.  granulated 

sugar  @      .10 

128  lb.  coffee 

@      .43f 

2.  Feb.  1,  A.  W.  Smith  &  Co.,  Boston,  Mass.,  sold  to 
Jones  &  French,  Marblehead,  Mass.,  on  30  days'  credit : 
36  boxes  oranges  at  $3.66| ;  12  chests  T.  H.  tea,  840  lb., 
at  50e;  14  chests  Japan  tea,  980  lb.,  at  37^^;  12  bbl. 
St.  Louis  flour  at  $9.10  ;  4  bags  coffee,  576  lb.,  at  33^^ ; 
54  boxes  lemons  at  $4.60  ;  14  bbl.  pineapples  at  $7.50  ; 
44  bunches  bananas  at  $5.45.     Write  the  bill. 

3.  R.  C.  Adams,  Danvers,  Mass.,  bought  of  Davis  & 
Bicknell,  Salem,  Mass.,  on  account  60  days:    25  bbl. 


RATIONAL  ARITHMETIC  31 

St.  Louis  flour  at  $11.48 ;  35  boxes  apricots,  25  lb.  each, 
at  23^  ;  20  boxes  apples,  25  lb.  each,  at  114^ ;  10  boxes 
peaches,  25  lb.  each,  at  374|Z^ ;  6  boxes  raisins,  25  lb. 
each,  at  25^;  8  boxes  prunes,  25  lb.  each,  at  16f^; 
13  boxes  currants,  25  lb.  each,  at  16^ ;  15  cases  Quaker 
Oats  at  $3.75  ;  12  cases  canned  corn,  24  doz.,  at  $2.12| ; 
18  cases  canned  tomatoes,  36  doz.,  at  $1.87^;  15  chests 
Japan  tea,  70  lb.  each,  at  54^;  1600  lb.  gunpowder 
tea  at  43i^.     Write  the  bill. 

4.  Jan.  31,  K.  R.  Good  &  Co.  sold  to  Lewis  W.  Sears, 
Middleton,  Mass.,  on  60  days'  time  :  6  pairs  men's  kid 
gloves  at  $2.48 ;  5  doz.  napkins  at  $5.50 ;  4  doz. 
children's  hose  at  $2.75 ;  9  pr.  blankets  at  $6.50 ; 
2  pieces  jeans,  80  yd.,  at  25f^;  2  pieces  point,  80  yd., 
at  12^^ ;  5  doz.  towels  at  $3.50 ;  15  doz.  spools  thread 
at  87i^ ;  7  doz.  ladies'  collars  at  $2.12^;  7  doz.  ladies' 
cuffs  at  $3.25 ;  14  robes  at  $2.33i ;  4  pieces  Irish 
linen,  156  yd.,  at  66f  ^.     Write  the  bill. 

5.  March  1,  James  K.  Broderick  &  Co.,  Boston, 
Mass.,  sold  to  Henry  T.  Lewis,  Peabody,  Mass.,  on 
30  days'  time:  2  pieces  gingham,  63  yd.,  at  37^^; 
1  piece  blue  denim,  31  yd.,  at  28^;  1  piece  brown 
denim,  30  yd.,  at  28^^ ;  1  piece  duck,  31  yd.,  at  50<z^ ; 
1  piece  shirting,  35  yd.,  at  22f  ^ ;  1  piece  sheeting, 
29  yd.,  at  43|^ ;  2  pieces  cottonnade,  76  yd.,  at  33^^; 
1  piece  jeans,  34  yd.,  at  30^  ;  1  piece  Irish  linen,  40  yd., 
at  62^^;  2  pieces  jaconet,  84  yd.,  at  25^;  5  doz. 
children's  hose  at  $2.75 ;  8  pr.  ladies'  kid  gloves  at 
$2.50  ;  4  doz.  spools  thread  at  62^^^.     W^ite  the  bill. 


PERCENTAGE 

Study  242  to  254  inclusive. 
Reference  244. 

32.    Express    the    following,    both    in    decimal    and 
common  fractional  forms : 


1. 

H%. 

9. 

38%. 

17. 

137i%. 

24. 

3  9C7 

4  0  /O* 

2. 

2t%- 

10. 

431%. 

18. 

1431%. 

25. 

15C/ 
16  /O* 

3. 

3i%. 

11. 

681%. 

19. 

156i%. 

26. 

.2%. 

4. 

31%. 

12. 

92i%. 

20. 

162i%. 

27. 

.4%. 

5. 

7i%. 

13. 

931%. 

21. 

1%. 

28. 

.5%. 

6. 

7f%. 

14. 

112^%. 

22. 

fV%. 

29. 

.12% 

7. 

3H%. 

15. 

1181%. 

23. 

25/0' 

30. 

.25% 

8. 

O/C-^  /q' 

16. 

13H%. 

References  255,  256,  257,  258. 


33.  Find: 

1.  23%  of  460  acres. 

2.  34%  of  60  cords. 

3.  15%  of  $75. 

4.  55%  of  280  feet. 

5.  45%  of  360. 

6.  62i%  of  $.50. 

7.  48%  of  175  gallons. 


8.  77%  of  430  bushels. 

9.  85%  of  1270  pounds. 

10.  1661%  of  480  tons. 

11.  225%  of  699  bushels. 

12.  625%  of  $4560. 

13.  .2%  of  $1750. 

14.  .4%  of  $1825. 


32 


RATIONAL  ARITHMETIC  33 

15.  .6%  of  $156.25.  21.  6|%  of  lUh 

16.  i%  of  $86,424.  22.  7^%  of  87^. 

17.  3.5%  of  $1250.  23.  |%  of  $1260. 

18.  4.4%  of  $875.  24.  36|%  of  $1214. 

19.  7i%  of  $1560.  25.  |%  of  33i  gallons. 

20.  6i%  of  8|. 

26.  A  man  having  $1960  spent  23%o  of  it.  How 
much  did  he  have  left  ? 

27.  A  gentleman  having  $^5,5Q5  invested  18% 
of  it  in  city  lots,  22%  in  railroad  stock,  30%  of  it  in 
bank  stock,  and  the  rest  in  a  truck  farm.  How  much 
did  he  invest  in  each  kind  of  property  ? 

28.  13%  of  a  grocer's  bill  of  $1665  was  for  coffee  at 
45^  per  pound.     How  many  pounds  did  the  grocer  buy  ? 

29.  I  bought  200  little  pigs  at  $7.50  each ;  25%  of 
them  died.  At  what  price  per  head  must  I  sell  those 
that  are  left  in  order  to  incur  no  loss  ? 

30.  A  farmer  who  raised  1880  bushels  of  corn  sold 
37i%  of  it  at  87^^  a  bushel.  How  much  did  he  receive 
for  what  was  sold  ? 

31.  An  agent  collected  $2430  for  his  client  whom  he 
charged  a  5%  fee  for  his  services.  How  much  did  he 
receive  and  how  much  did  he  remit  to  his  client  ? 

32.  C.  E.  Bates  has  failed  owing  me  $1645.  He  is 
able  to  pay  only  43%  of  his  debts.  At  that  rate,  how 
much  should  I  receive  ? 

33.  I  bought  27  bales  of  cloth,  12  pieces  to  the  bale 
averaging  47  yards  to  the  piece,  at  $1.25  per  yard. 
For  what  must  I  sell  the  entire  quantity  to  gain  16f  %  ? 


34  RATIONAL  ARITHMETIC 

34.  A  private  bank  that  has  failed  declared  a 
dividend  of  87i%.  A's  balance  on  deposit  was 
$6,437.50,  B's  $3,856.56,  C's  $872.  How  much  did 
each  lose  ? 

35.  A  man,  at  his  death,  left  an  estate  valued  at 
$150,000.  He  left  10%  of  it  to  organized  charity,  12^% 
of  it  to  his  college,  and  4i%  to  his  church.  He  divided 
the  remainder  among  his  family  so  that  his  wife  received 
62^%  of  it  and  his  son  and  daughter  each  18f  %.  What 
did  each  receive  ? 

References  ^259,  260,  261. 

34.  1.  18  is  9%  of  what.^ 

2.  38  is  5%  of  what  ? 

3.  54  is  12i%  of  what  ? 

4.  84  is  15%  of  vv^hat  ? 

5.  $460  is  23%  of  what .? 

6.  $1143  is  35%  of  what. ^ 

7.  $650  is  42%o  of  what  ? 

8.  $9420  is  96%  of  what  ? 

9.  $3150  is  3.5%  of  what? 

10.  $48.60  is  7.5%  of  what. ^ 

11.  $148.20  is  16|%  of  what.? 

12.  $1375.50  is  33i%  of  what.? 

13.  $1198  is  55.5%  of  what? 

14.  $6570  is  234%  of  what  ? 

15.  $1254  is  675%  of  what? 

16.  Of  my  flock  of  pigeons  16|%  died.  If  215  are 
left,  how  many  pigeons  were  there  in  the  original  flock  ? 


RATIONAL  ARITHMETIC  35 

17.  I  have  just  paid  a  bill  of  $135.45,  which  repre- 
sented 18f%  of  my  available  cash.  How  much  did  I 
have  before  paying  the  bill  ?     How  much  is  left  ? 

18.  A  broker  received  $^6.25  as  his  fee  for  selling  a 
certain  piece  of  property  on  a  commission  of  2^%. 
What  was  the  value  of  the  property  ? 

19.  An  analvsis  shows  that  a  merchant's  costs  and 
fixed  charges  amounted  to  87^%  of  his  gross  sales  for  a 
certain  year.  If  his  total  expenses  and  costs  amounted 
to  $246,400,  what  were  his  sales  ? 

20.  A  merchant  sold  15%  of  his  stock  of  goods  for 
$45,350.  What  was  his  entire  stock  worth  before  he 
sold  any  ? 

21.  A  farmer  bought  87  acres  of  land  which  is  37^% 
of  what  he  previously  owned.  How  much  did  he  own 
after  the  purchase  ? 

22.  A  sold  a  carriage  at  a  profit  of  16f%,  thereby 
gaining  $43.25.  What  did  it  cost  and  what  did  it 
sell  for? 

23.  If  it  takes  60  days  to  complete  16f%  of  a  con- 
tract, how  long  will  it  take  to  finish  the  job  ? 

References  262,  263. 

35.     1.    What  per  cent  of  260  is  13  ? 

2.  What  per  cent  of  480  is  72  ? 

3.  What  per  cent  of  $2.40  is  $32  ? 

4.  What  per  cent  of  $188.50  is  $22.62.^ 

5.  What  per  cent  of  $640  is  $131.60  ? 

6.  What  per  cent  of  28  bu.  is  7  bu.  ? 


36  RATIONAL   ARITHMETIC 

7.  What  per  cent  is  $314.50  of  $1850  ? 

8.  What  per  cent  of  $.95  is  $.70.^ 

9.  What  per  cent  of  $2664  is  $826.04  ? 


10.    What  per  cent  of  1^  is 


3.5 

8  • 


11.  What  per  cent  of  77|  is  66f  ? 

12.  What  per  cent  of  $456.75  is  $219.24? 

13.  W^hat  per  cent  is  $1414.80  of  $5240? 

14.  What  per  cent  of  324  is  64.8  ? 

15.  What  per  cent  of  $1940  is  $9.70? 

16.  A  man  owing  a  debt  of  $1680,  paid  $940.80. 
What  per  cent  remains  unpaid  ? 

17.  In  a  school  of  480  pupils,  24  were  absent  on  a 
certain  day.     What   was  the  percentage  of  absence? 

18.  A  merchant  purchased  goods  for  $425  and  sold 
them  for  $510.  How  many  dollars  did  he  gain?  This 
was  what  per  cent  of  the  cost?  What  per  cent  of  the 
selling  price  ? 

19.  A  lawyer  charged  $14.19  for  collecting  a  claim 
of  $473.     What  rate  per  cent  did  he  charge? 

20.  I  sell  a  house  that  cost  me  $2500  for  $2125. 
The  loss  is  what  per  cent  of  the  cost  ? 

21.  Of  an  army  of  45,000  men  5625  were  killed  in 
battle  and  10,125  were  wounded.  What  was  the  per- 
centage of  loss  ? 

22.  An  insurance  company  with  a  capital  stock  of 
$250,000  declared  an  annual  dividend  of  $21,250. 
The   dividend    was    what   percentage   of   the   capital? 

23.  A  miller  keeps  one  quart  out  of  every  bushel 
he  grinds.     What  is  the  percentage  of  his  toll  ? 


RATIONAL  ARITHMETIC  37 

24.  A  merchant  invested  $34,395.30  in  business. 
At  the  end  of  the  year  he  finds  he  ha:  gained  $3821.70. 
This  gain  is  what  per  cent  of  the  money  invested  ? 

25.  If  I  sell  f  of  a  quantity  of  goods  for  what  f  of 
them  cost,  what  is  the  gain  per  cent  ? 

26.  John  Brown,  failing  in  business,  owes  $3650. 
His  entire  resources  are  $2920.  What  per  cent  of  his 
indebtedness  can  he  pay  ? 

27.  The  assets  of  an  insolvent  concern  are  $23,450 ; 
its  liabilities  are  $33,500.  What  per  cent  can  it  pay 
and  what  will  A  receive  on  a  claim  of  $1350  ? 

28.  I  paid  $300  for  apples  bought  at  $5.40  a  barrel. 
I  sold  38  barrels  for  $220.40.  What  percentage  did  I 
gain  on  the  quantity  sold  ? 

29.  The  enrollment  in  a  certain  High  School  is  846. 
102  are  enrolled  in  the  Classical  Course,  228,  the 
General  Course,  246,  the  Manual  Training  Course, 
and  270,  the  Commercial  Course.  What  percentage 
of  the  school  could  each  course  claim  .^^  Carry  the 
result  to  the  fourth  decimal  place  if  necessary. 

30.  At  the  close  of  the  baseball  season  of  1915,  the 
first  four  teams  in  the  American  League  stood  as 
given  below.  Figure  the  percentage  of  each,  carrying 
to  the  fourth  decimal  place. 

WON  LOST 


Boston       .     .     .     . 

101 

50 

Detroit      .     .     .     . 

100 

54 

Chicago     .... 

93 

61 

Washington    .     .     . 

85 

68 

38  RATIONAL  ARITHMETIC 

GENERAL   PROBLEMS   IN   PERCENTAGE 

The  following  problems  cover  the  entire  range  of 
simple  percentage.  A  thorough  understanding  of  the 
matter  covered  by  paragraphs  '24'2  to  263  inclusive 
will  enable  the  student  to  solve  them  with  facility 
and  accuracv. 

These  problems  are  not  graded  so  as  to  present 
increasing  difhculties.  They  are  rather  arranged  so 
an  average  lesson  of  ten  problems  will  offer  the  varying 
conditions  of  ease  and  difficult}^  that  are  usually  found 
in  actual  experience. 

Every  problem  should  be  solved  in  the  easiest 
possible  way. 

36.  1.  After  experiencing  a  loss  of  $5000,  a  business 
man  has  $30,000  left.  His  loss  was  what  per  cent  of 
his  original  capital.^ 

2.  The  assets  of  a  bankrupt  were  $27,179.38.  His 
liabilities  were  $43,487.  What  per  cent  could  he  pay  ? 
How  much  would  be  due  A  whose  claim  is  $3540.75  ? 

3.  One  of  our  creditors  who  met  with  financial 
reverses  agreed  to  pay  our  claim  in  annual  install- 
ments of  16f%.  He  has  made  four  payments.  How 
much  does  he  still  owe,  the  original  claim  being 
$1847.38? 

4.  A  owns  i  interest  in  a  business,  B  f,  and  C  J. 
A  sells  33i%  of  his  share  for  $1874.  What  is  the  entire 
value  of  the  business?  What  is  B's  share?  What  is 
C's  share? 


RATIONAL  ARITHMETIC  39 

5.  A  and  B  are  partners.  A's  investment  is 
$36,783.60  and  B's  is  $26,636.40.  To  what  percentage 
of  the  profits  is  each  entitled  ? 

6.  Out  of  an  inheritance  of  $17,500,  I  invested  45% 
in  real  estate,  25%  in  United  States  bonds,  and  put 
the  rest  into  a  mortgage.  What  sum  does  the  mortgage 
represent  ? 

7.  A  owTis  i  of  the  stock  of  a  corporation,  B  25%, 
C  i,  D  37i%,  and  E  the  remainder.  B's  investment 
is  $3500.     What  is  E's  investment  ? 

8.  A  and  B  engage  in  business  as  partners.  A 
invests  $6980  and  B  $5584.  Each  partner's  share 
represents  what  per  cent  of  the  total  investment? 

9.  A  regiment  went  into  battle  with  938  men  and 
came  out  with  804.     What  percentage  was  lost.^^ 

10.  Aly  holdings  in  real  estate  are  worth  $8500 ;  my 
personal  property  is  worth  $4350.  If,  during  the 
coming  year,  my  real  estate  increases  in  value  23% 
and  my  personal  property  9%,  what  will  be  the  total 
value  then  and  what  will  be  the  percentage  of  increase 
of  both  ? 

11.  An  operator  bought  a  large  tract  of  land  and 
sold  40%  of  it  to  one  customer,  20%  of  the  remainder 
to  a  second  customer,  25%  of  what  still  remained  to  a 
third  customer.  If  234  acres  remain  unsold,  how  many 
acres  were  there  in  the  original  tract  of  land  .^ 

12.  In  1918  a  merchant's  sales  amounted  to  $234,520. 
In  1919  they  amounted  to  $213,413.20.  If  they 
increase  during  1920  at  the  same  rate  that  they  de- 
creased in  1919,  what  will  be  the  sales  for  1920^ 


40  RATIONAL  ARITHMETIC 

13.  On  January  17,  1916,  merchandise  was  bought 
for  $1475.85  on  three  months'  credit,  subject  to  a  dis- 
count of  3%  if  paid  within  ten  days.  What  sum  was 
required  to  settle  the  bill  on  January  27,  1916.^ 

14.  Two  railroads,  one  300  miles  long  and  the  other 
500  miles  long,  carry  340  barrels  of  potatoes  at  a 
through  rate  of  23^  a  barrel.  This  freight  is  to  be 
divided  in  proportion  to  each  railroad's  per  cent  of  the 
total  mileage.     How  much  does  each  road  receive? 

15.  An  agent  sold  160  bales  of  cotton,  averaging 
240  lb.  each,  at  32i^  a  pound;  75  hhd.  of  tobacco, 
averaging  360  lb.  each,  at  26f  ^  a  pound ;  130  bbl. 
sugar,  averaging  180  lb.  each,  at  10^^  a  pound.  He 
charges  ^i%  of  the  amount  received.  What  was  his 
charge  ? 

16.  A  speculator  bought  a  farm  of  175  acres  of  land 
for  $11,900  and  sold  64  acres  for  $5440.  What  per 
cent  did  he  gain  on  the  part  sold  ? 

17.  By  energetic  effort,  the  sales  department  of  a 
certain  business  was  able  to  increase  the  sales  20%  each 
year  for  three  successive  years.  The  total  increase 
amounted  to  $36,400.  What  were  the  sales  the  year 
before  the  first  increase  was  effected  ? 

18.  A  9%  dividend  on  stock  amounted  to  $873. 
What  was  the  face  value  of  the  stock  ? 

19.  A  second-hand  dealer  sold  two  automobiles  for 
$640  each.  On  one  he  gained  20%  and  on  the  other  he 
lost  20%.  Did  he  gain  or  lose  by  the  transaction? 
How  much? 


RATIONAL  ARITHMETIC  41 

20.  A  speculator  invested  $5340.  He  gained  10% 
the  first  year,  13%  the  second  year,  lost  18%  the  third 
year,  and  gained  5%  the  fourth  year.  What  is  his 
capital  at  the  end  of  the  fourth  year  ? 

21.  Several  years  ago  176,834,300  pounds  of  fish 
passed  through  the  Boston  market.  Of  this  quantity, 
Gloucester  furnished  110,637,829  pounds.  At  a  more 
recent  date  the  total  amounted  to  215,643,330  pounds, 
of  which  Gloucester  furnished  120,418,563  pounds. 
What  per  cent  was  furnished  by  Gloucester  in  each 
year? 

22.  An  automobile  manufacturer  decided  to  reduce 
the  price  of  his  cars  10%,  and  called  upon  his  sales 
department  to  increase  the  sales  a  sufficient  amount  to 
counteract  the  reduction  in  price.  What  per  cent 
increase  would  the  general  manager  of  the  sales  depart- 
ment be  obliged  to  show  ? 

23.  A  certain  piece  of  property  having  depreciated 
$2355,  is  now  worth  $3925.  What  was  its  original 
value  ?     What  per  cent  has  it  depreciated  ? 

24.  A  leather  manufacturer  owning  75%  of  a  factory 
building  sold  16f%  of  his  share  and  received  $1555. 
Find  the  value  of  the  factory. 

25.  Hoyt's  stock  of  goods  is  worth  $9462,  which  is 
15%  more  than  Taylor's,  and  15%  less  than  Ashton's. 
What  is  the  value  of  the  stock  carried  by  each  ? 


PROFIT  AND  LOSS 

Study  265  to  273  inclusive. 
References  274,  275. 

37.  What  is  the  profit  or  loss  and  seUing  price  of 
goods  costing : 

1.  $43.42  and  sold  at  a  profit  of  9%  ? 

2,  $87.54  and  sold  at  a  profit  of  12^%  ? 
I.    $175.89  and  sold  at  a  profit  of  27%  ? 

4.  $21.43  and  sold  at  a  loss  of  8i%? 

5.  $312.51  and  sold  at  a  loss  of  13f%? 

6.  $15.07  and  sold  at  a  profit  of  371%  ? 

7.  $102.73  and  sold  at  a  profit  of  SSi%  ? 

8.  $240.81  and  sold  at  a  loss  of  15%.^ 

9.  $181.03  and  sold  at  a  profit  of  28|%)? 

10.  $37.56  and  sold  at  a  profit  of  20%? 

11.  Property  valued  at  $3750  increased  8^%  in 
value.  It  was  then  sold.  What  was  the  profit? 
What  was  the  selling  price  ? 

12.  If  I  buy  cloth  at  $5.40  and  sell  it  at  161%,  loss, 
what  is  my  selling  price  ? 

13.  Bought  an  automobile  for  $1145.  It  was  then 
sold  at  a  loss  of  16|%.  What  was  the  loss  and  what  was 
the  selling  price  ? 

14.  Paid  $240  for  a  pair  of  horses  and  sold  them 
at  a  profit  of  31^%.     How  much  did  I  gain? 

42 


RATIONAL  ARITHMETIC  48 

15.  Bought  hats  for  $36  a  dozen  and  sold  them  at 
a  profit  of  33^%.  What  was  the  seUing  price  of  each 
hat? 

16.  My  balance  sheet  shows  that  advertising,  rent, 
clerk  hire,  etc.,  commonly  called  overhead  charges , 
amount  to  about  8^%  of  the  total  amount  of  my  pur- 
chases. To  leave  a  proper  margin  of  safety  I  have 
decided  to  figure  these  overhead  charges  as  10%  of 
the  cost.  In  order  to  provide  for  this,  how  much 
must  we  mark  goods  costing  $13.40  to  clear  a  profit 
of  15%  ? 

17.  Allowing  an  overhead  of  10%,  what  must  the 
following  goods  be  marked  to  show  a  profit  of  10%? 

of  15%  ?   of  20%  ? 

Refrigerators  costing  $  24.50 

Chamber  sets  costing  125.80 

Persian  rugs  costing  163.49 

Pianos  costing  560. 

Buffets  costing  61.18 

Dining-room  sets  costing  242.30 

Tea- wagons  costing  21.73 

Couches  costing  75.50 

Parlor  sets  costing  407.92 

Veranda  sets  costing  195.60 

References  276,  277,  278. 

38.  Find  the  cost : 

1.  Loss  $151.20,  rate  of  loss  10%. 

2.  Loss  $107.91,  rate  of  loss  2^%. 

3.  Loss  $205.78,  rate  of  loss  li%. 


44 


RATIONAL   ARITHMETIC 


4.  Loss  $456.38,  rate  of  loss  2^%. 

5.  Loss  $220.15,  rate  of  loss  6i%. 

6.  Loss  $117.50,  rate  of  loss  8^%. 

7.  Gain  $34.23,  rate  of  gain  21%. 

8.  Gain  $79.85,  rate  of  gain  25%. 

9.  Gain  $12.73,  rate  of  gain  33i%. 

10.  Gain  $19.05,  rate  of  gain  22|%. 

11.  Gain  $22,  rate  of  gain  16^%. 

12.  Gain  $17.98,  rate  of  gain  20%. 

13.  Selling  price  $1056.80,  rate  of  gain  5%. 

14.  Selling  price  $2435.28,  rate  of  gain  20%. 

15.  Selling  price  $3672.25,  rate  of  gain  16|%. 

16.  Selling  price  $1434.75,  rate  of  gain  12^%. 

17.  Selling  price  $1806.75,  rate  of  gain  33^%. 

18.  Selling  price  $2584.82,  rate  of  gain  2%. 

19.  Selling  price  $950.28,  rate  of  loss  5%. 

20.  Selling  price  $42.00,  rate  of  loss  50%. 

21.  Selling  price  $245.75,  rate  of  loss  16f%. 

22.  Selling  price  $5042.80,  rate  of  loss  22|%. 

23.  Selling  price  $550.25,  rate  of  loss  6|%. 

24.  Selling  price  $64.80,  rate  of  loss  1|%. 

25.  By  selling  goods  for  $140  I  lose  12i%.  At  what 
price  must  I  sell  them  to  gain  12^%  ? 

26.  Sold  a  row  boat  that  cost  $80  at  a  gain  of  12^%, 
and  with  the  proceeds  I  purchased  another  boat  which 
I  sold  at  a  loss  of  10%.  How  much  did  I  gain  by  both 
transactions  ? 

27.  A  cow  was  sold  for  $67.50  which  was  10%  below 
cost.  What  was  the  cost  and  what  was  the  loss  ? 


RATIONAL  ARITHMETIC  45 

28.  I  bought  a  safe  for  $118.80.  This  was  10% 
higher  than  the  manufacturer's  price.  What  was  the 
retailer's  profits  ? 

29.  A  merchant  marked  his  goods  at  37^%  above 
cost.  What  is  the  cost  of  an  article  that  he  marked 
at  $156.64.^ 

30.  I  sold  goods  to  a  customer  at  a  price  which  would 
have  netted  me  a  profit  of  *'25%  had  the  customer  paid 
his  bill.  He  failed,  however,  and  was  able  to  pay  me 
only  S5(ji  on  the  dollar.  In  spite  of  this,  I  netted  a 
profit  of  $314  on  the  transaction.  What  was  the 
amount  of  my  bill  ? 

31.  A  manufacturer  gained  25%;  the  wholesaler 
made  a  profit  of  20% ;  the  retailer  made  a  profit  of 
33^%.  What  was  the  actual  manufacturing  cost  of 
an  article  the  retail  price  of  w^hich  was  $100  .^^ 

32.  Suppose,  in  the  above  question,  that  the  manu- 
facturer should  sell  direct  to  the  customer,  thereby 
increasing  the  cost  of  the  article  10%.  At  what  price 
could  he  afford  to  sell  at  retail  and  still  make  the  same 
per  cent  profit  that  he  now  enjoys  .^^ 

References  279,  280. 
39.    Find  the  per  cent  of  gain  or  loss : 

1.  Cost  $105.50,  gain  $21.10. 

2.  Cost  $165,  gain  $49.54. 

3.  Cost  $40.75,  gain  $7.34. 

4.  Cost  $140.75,  gain  $14.08. 

5.  Cost  $200,  gain  $75.50. 


46  RATIONAL  ARITHMETIC 

6.  Cost  $103.40,  loss  $17.23. 

7.  Cost  $755.90,  loss  $60.47. 

8.  Cost  $64.80,  loss  $19.44. 

9.  Cost  $21.70,  loss  $1.74. 
10.  Cost  $75,  loss  $23.25. 
^11.  Cost  $220,  gain  $125.40. 

12.  Cost  $115.20,  gain  $63.36. 

13.  Cost  $1256.40,  gain  $527.68. 

14.  Cost  $750.48,  gain  $202.63. 

15.  Cost  $90,  gain  $14.85. 

16.  Cost  $15.24,  gain  $8.08. 

17.  Cost  $420.50,  gain  $50.47. 

18.  Cost  $50.17,  gain  $9.03. 

19.  Cost  $430.85,  gain  $12.93. 

20.  Cost  $59.84,  loss  $17.95. 

21.  Cost  $2412.50,  loss  $627.25. 

22.  Cost  $650.75,  loss  $221.26. 

23.  Cost  $108,  loss  $54. 

24.  Cost  $542,  loss  $338.75. 

25.  A  milliner  bought  hats  at  $27  a  dozen  and  sold 
them  for  $3  each.     What  was  the  gain  per  cent  ? 

26.  What  is  gained  by  buj^ing  paper  at  $2  a  ream 
and  retailing  it  for  1  ^  a  sheet  ? 

27.  I  bought  a  horse  for  $350  and  sold  him  for  $400. 
What  was  the  gain  per  cent.'^ 

28.  If   a   horse   cost   $400   and    was   sold   for    $350, 
what  per  cent  was  the  loss  .^ 


RATIONAL  ARITHMETIC  47 

29.  A  stationer  bought  2  bundles  of  paper  for  $1.75 
a  ream  and  sold  it  at  retail  at  the  rate  of  3  sheets  for  2^. 
What  per  cent  did  he  gain  and  how  much  did  he  gain 
in  aW? 

30.  A  dealer  bought  150  crates  of  fruit  for  $1  a 
crate.  He  sold  35  crates  at  $1.25  a  crate,  30  crates 
at  $1.15  a  crate,  60  crates  at  $1.20  a  crate,  10  crates 
at  cost,  and  threw  the  rest  away  as  worthless.  What 
did  he  gain  or  lose  and  what  per  cent  ? 

31.  What  per  cent  profit  will  be  realized  from  the 
sale  of  peaches  at  3^  each  if  they  cost  $1.25  a  hundred 
and  10%  of  them  are  lost  by  decay  ? 

32.  I  sold  C.  S.  Chase  goods  that  cost  $430  so  as  to 
make  a  profit  of  60%,  on  30  days'  credit.  Before  the 
account  was  due,  Chase  failed,  paying  only  37^^  on 
the  dollar.     What  was  my  per  cent  of  loss  ? 

33.  A  merchant  bought  gloves  at  $8  per  dozen 
pairs  and  sold  them  at  $1.25  a  pair.  What  was  the 
per  cent  gained  ? 

GENERAL    PROBLEMS   IN    PROFIT   AND   LOSS 
References  274  to  "280  inclusive. 

40.  1.  By  selling  goods  at  a  profit  of  37^%  I  made 
$215.45.  What  do  the  goods  cost  and  what  do  they 
sell  for  ? 

2.  Goods  were  sold  at  a  profit  of  20%.  If  the  seller 
received  $36.54,  what  was  his  profit.^ 

3.  I  bought  tea  for  53^  a  pound.  What  price  must 
I  sell  it  for  in  order  to  gain  32%  ? 


48  RATIONAL  ARITHMETIC 

4.  What  is  the  loss  on  goods  sold  for  $4348.50, 
which  is  19%  below  cost? 

5.  Goods  are  sold  for  $436.28  at  a  loss  of  15%. 
For  what  price  should  they  be  sold  to  gain  15%  ? 

6.  A  dealer  sold  hats  at  retail  for  $3.50  each,  and 
at  wholesale  for  $33  a  dozen.  At  retail,  his  profit  was 
40%.  Does  the  wholesale  price  show  a  profit  or  loss.^^ 
How  much  per  hat  ?     What  per  cent  "^ 

7.  A  barrel  of  flour  sold  for  $9.45  nets  a  profit  of 
35%.  At  what  price  could  we  sell  it  if  we  were  content 
with  a  profit  of  20% .? 

8.  A  coal  dealer  buys  coal  for  $9.75  by  the  long  ton 
and  sells  it  for  $11.75  by  the  short  ton.  What  per  cent 
profit  does  he  make  ? 

9.  A  sold  a  factory  building  to  B  for  $8621.  By 
so  doing  he  lost  10%.  B  expended  $2300  installing 
a  sprinkler  system  and  then  sold  the  factory  for  20% 
more  than  A  paid  for  it.  How  much  did  B  gain  and 
what  per  cent  did  he  gain  on  his  investment  ? 

10.  How  much  should  be  asked  a  pound  for  fish 
costing  $6.50  a  hundred  to  net  a  profit  of  10%  and 
allow  for  10%  waste? 

11.  A  book  agent  sells  two  books  for  $5  each.  On 
one  he  loses  20%,  and  on  the  other  he  gains  20%.  Does 
he  gain  or  lose,  how  much  and  what  per  cent  ? 

12.  Goods  bought  at  $4  a  gross  and  sold  at  40^  a 
dozen  yield  what  per  cent  profit  ? 

13.  I  sold  goods  to  a  retailer  at  a  profit  of  40%. 
Before  settlement  he  failed,  paying  25^  on  the  dollar. 
What  was  my  loss  on  goods  costing  $320  ? 


RATIONAL  ARITHMETIC  49 

14.  Bought  a  horse  from  A  at  20%  less  than  it  cost 
him.  Sold  it  for  25%  more  than  I  paid  for  it.  I 
gained  $15  in  the  transaction.  What  did  the  horse 
cost  A ;  what  did  it  cost  me ;  how  much  did  I  sell  it  for  ? 

15.  I  buy  oranges  at  $2.50  a  hundred.  What  price 
must  I  mark  them  a  dozen  to  gain  25%,  allowing  10% 
for  decay  and  15%  overhead  expenses  ? 

16.  A  dealer  buys  6  bags  Rio  coffee,  218  lb.  in  a 
bag,  at  4 If  ^  a  pound ;  12  bags  Java  coffee,  75  lb.  in  a 
bag,  at  25^  a  pound.  After  mixing  the  two  kinds,  he 
sells  at  a  profit  of  75%.      What  is  the  price  a  pound  ? 

17.  What  price  can  I  afford  to  pay  for  property 
that  rents  for  $50  a  month  in  order  to  make  a  net 
profit  of  5%  a  year,  allowing  $200  annually  for  neces- 
sary repairs,  taxes,  etc.  .^ 

18.  Find  the  gain  or  loss  per  cent  of  each  of  the 
following,  carrying  to  the  fourth  decimal  place : 

A.  Purchases $13,502.10 

Sales 12,786.50 

Inventory  at  closing      .     .     .         4,983.70 

B.  Inventory  at  beginning      .     .  3,908.00 

Purchases 10,680.20 

Sales 10,450.50 

Inventory  at  closing      .     .     .  6,400.75 

C.  Inventory  at  beginning      .     .  3,100.85 

Purchases    , 8,900.50 

Sales 11,500.00 

Returned  to  us 540.30 

Returned  by  us 560.50 

Inventory  at  closing      .     .     .  4,550.00 


I 


TRADE  DISCOUNT 

Study  284  to  290  inclusive. 
References  291,  292. 

41.    Find  the  net  amount  of  the  following  bills : 

1.  List  price  $340.25,  discount  35%. 

2.  List  price  $1256.35,  discount  27%. 

3.  List  price  $438.40,  discount  23%. 

4.  List  price  $750.50,  discount  33^%^. 

5.  List  price  $755.75,  discount  37^%. 

6.  List  price  $351.20,  discount  20%^. 

7.  List  price  $1050.30,  discount  16%. 

8.  List  price  $978.80,  discount  42%. 

9.  List  price  $127.70,  discount  21%. 

10.  List  price  $2040.50,  discount  2%. 

11.  List  price  $1434.25,  discounts  20%  and  10%o. 

12.  List  price  $760.20,  discounts  12i  %  and  10%. 

13.  List  price  $126.34,  discounts  20%o  and  25%. 

14.  List  price  $285.40,  discounts  27i%  and  20%. 

15.  List  price  $1244.18,  discounts  25%  and  15%. 

16.  List  price  $556.30,  discounts  27%)  and  12%). 

17.  List  price  $112.80,  discounts  17%  and  10%. 

18.  List  price  $680.12,  discounts  Q^%  and  5%.^ 

19.  List  price  $120.40,  discounts  22%  and  20%). 

50 


RATIONAL  ARITHMETIC  51 

20.  List  price  $450.75,  discounts  16%  and  12%. 

21.  List  price  $845.12,  discounts  20%,  10%,  and  5%. 

22.  List  price  $360.70,  discounts  W%,  25%,  and  10%. 

23.  List  price  $850.20,  discounts  25%,  25%,  and  10%. 

24.  List  price  $351.65,  discounts  10%,  10%,  and  5%. 

25.  List  price  $1201.30,  discounts  18%,  7%,  and  3%. 

26.  List  price  $700.50,  discounts   16f%,  ni%,  and 
6i%. 

27.  List  price  $325.40,  discounts  12%,  10%,  and  5%. 

28.  List  price  $970.60,  discounts  15%,  12%,  and  4%. 

29.  List  price  $1010.25,  discounts  15%,  10%,  and  8%. 

30.  List  price  $242.50,  discounts  25%,  25%o,  and  5%. 

31.  List  price  $1068.20,  discounts  50%,  50%,  and  10%. 

32.  List  price  $978.35,  discounts  25%,  40%,  and  35%. 

33.  List  price  $326.30,  discounts  50%^,  30%,  and  25%. 

34.  List  price  $829.20,  discounts  40%,  50%,  and  10%. 

35.  List  price  $2048.80,  discounts    60%,    25%o,    and 
15%. 

36.  List  price  $1218.75,  discounts  62^%,  37^%^,  and 
10%. 

37.  List   price    $650.50,  discounts    66|%,  40%,  and 
H%. 

38.  List  price  $450.20,  discounts  50%o,  30%,  and  20%). 

39.  List  price  $360.20,  discounts  65%o,  15%o,  and  10%). 

40.  List  price  $520.80,  discounts  30%,  50%,  and  20%. 


52  RATIONAL  ARITHMETIC 

Reference  293. 

42.  What  single   discount  is    equal   to  a   discount 
series  of : 

1.  20%,  10%,  and  5%?  11.    25%,  15%,  and  Q%? 

2.  30%,  10%,  and  5%?  12.    50%,  25%,  and  10%.?^ 

3.  10%,  5%,  and  2%  ?  13.    35%,37i%,  and  12i%  ? 

4.  12%,  5%,  and  2%  ?  14.    40%,  25%,  and  10%  ? 

5.  20%,  8%,  and  5% .?  15.    16f%,  5%,  and  2%.? 

6.  15%,  10%,  and  6%?  16.    15%,  12%,  and  4%.^ 

7.  10%,  10%,  and  5%?  17.    30%,  10%,  and  8%? 

8.  20%,  12i%,  and  10%  ?  18.    10%,  5%,  and  10% .? 

9.  33i%,  10%,  and  10%?  19.    10%,  20%,  and  20%.^ 
10.  50%,  20%,  and  10%  ?  20.    25%,  25%,  and  10%  ? 

21.  50%,  50%o,  and  20%? 

22.  35%,  25%,  20%,  and  10%o? 

23.  50%,  20%,,  20%,  and  10%^  ? 

24.  60%,  40%,  and  20%  ? 

References  294,  295,  296. 

43.  Find  the  gross  amount  of  the  following  bills : 

1.  Discount  $125.30,  rate  of  discount  25%. 

2.  Discount  $104.15,  rate  of  discount  20%. 

3.  Discount  $275.50,  rate  of  discount  12^%. 

4.  Discount  $340.75,  rate  of  discount  5%. 

5.  Discount  $210.42,  rate  of  discount  ^IWc 

6.  Net  $134.80,  rate  of  discount  10%). 

7.  Net  $84.60,  rate  of  discount  60%. 

8.  Net  $234.50,  rate  of  discount  15%. 


RATIONAL  ARITHMETIC  53 

9.  Net  $119,  rate  of  discount  12^%. 

10.  Net  $218.^6,  rate  of  discount  30%. 

11.  Discount  $125,  rate  of  discount  35%. 

12.  Discount  $215.50,  rate  of  discount  25%, 

13.  Discount  $36.75,  rate  of  discount  12^%, 

14.  Discount  $105.15,  rate  of  discount  15%, 

15.  Discount  $341.70,  rate  of  discount  35%. 

16.  Net  $85.50,  rate  of  discount  25%,  10%o. 

17.  Net  $128.40,  rate  of  discount  40%,  25%,  10%. 

18.  Net  $275.75,  rate  of  discount  25%,  10%,  10%. 

19.  Net  $187.26,  rate  of  discount  50%,  10%,  5%. 

20.  Net  $150.50,  rate  of  discount  10%o,  5%,  2%. 

Reference  297. 
44.    What  rate  of  discount  shall  we  allow  on  a  bill  of  : 

1.  $346.40  to  net  $259.80? 

2.  $780  to  net  $530.40  ? 

3.  $112.50  to  net  $93.75? 

4.  $218.40  to  net  $136.50? 

5.  $187.26  to  net  $112.37? 

6.  $360  to  net  $288  ? 

7.  $450  to  net  $393.75? 

8.  $234.50  to  net  $195.42? 

9.  $312.90  to  net  $208.60? 

10.  $520  to  net  $374.40? 

11.  $760  to  net  $585.20? 

12.  $365.75  to  net  $219.45? 

13.  $1600  to  net  $1400? 


54  RATIONAL  ARITHMETIC 

14.  $2340  to  net  $1521  ? 

15.  $298  to  net  $172.84.^ 

16.  $169.75  to  net  $105.25.^ 

17.  $2359  to  net  $2064.13.^ 

18.  $250  to  net  $182.50? 

19.  $3296  to  net  $2307.20  .^ 

20.  $265.50  to  net  $199.13? 

Reference  298. 

45.  At  what  price  must  the  following  goods  be 
marked  in  order  to  allow  the  prescribed  discount  and 
still  make  the  designated  profit? 


Cost 

Required  Profit 

Discount 

1. 

$100 

37i% 

20% 

2. 

$250 

24% 

35% 

3. 

$304.75 

12% 

20%  and  10% 

4. 

$480.20 

35% 

'371% 

5. 

$1050.50 

40% 

25%,  10%,  and  5% 

6. 

$785.40 

33i% 

23% 

7. 

$570 

18% 

32% 

8. 

$921.20 

27% 

50% 

9. 

$893.68 

12i% 

25%  and  10% 

10. 

$723.85 

18% 

30%,  10%,  and  2% 

11. 

$5200 

22% 

21% 

12. 

$225.20 

50% 

10% 

13. 

$720.30 

27% 

5%  and  2% 

14. 

$850 

25% 

40% 

15. 

$365.50 

30% 

17% 

RATIONAL  ARITHMETIC  55 


Cost 

Required  Pkopit 

Discount 

16. 

$180 

40% 

10%,  10%,  and  59 

17. 

$127.50 

21% 

32% 

18. 

$150.25 

15% 

12i% 

19. 

$900 

10% 

5%,  5%,  and  10% 

20. 

$675.80 

20% 

23% 

GENERAL  PROBLEMS   IN  TRADE   DISCOUNT 
References  291  to  298  inclusive. 

46.  1.  I  bought  a  bill  of  $546.30  at  a  trade  discount 
of  20%,  10%,  and  5%  with  an  additional  cash  discount 
of  2%.  I  took  advantage  of  the  cash  rate  and  then  sold 
the  goods  at  the  original  list  price  with  a  flat  discount 
of  25%.     How  much  did  I  gain  ?     What  per  cent  ? 

2.  If  I  buy  goods  at  a  discount  of  30%  from  the  list 
price  and  sell  at  the  list  price,  what  per  cent  profit  do  I 
make  ? 

3.  Goods  bought  at  $6  a  gross  at  a  discount  of  20%, 
and  sold  at  75^  a  dozen  yield  what  per  cent  profit  .^^ 

4.  Which  is  better  and  how  much  on  a  bill  of  $240, 
a  discount  of  20%,  10%,  and  5%,  or  a  discount  of  35%? 

5.  By  selling  goods  at  $1.50  a  yard  I  make  25% 
profit.  What  must  I  mark  them  in  order  to  deduct 
10%  and  still  make  the  same  profit  .^^ 

6.  Goods  costing  $345  are  marked  up  30%  and  are 
then  sold  at  a  discount  of  20%.  How  much  is  gained 
and  what  is  the  gain  per  cent  ? 

7.  Goods  are  marked  up  20%.  What  discount  can 
the  seller  allow  on  this  price  and  still  net  the  cost.^ 


56  RATIONAL  ARITHMETIC 

Make  out  bills  for  the  following : 

8.  Use  current  date,  your  own  locality.  Adams 
and  Baxter  bought  of  the  Boston  Hardware  Company, 
terms  net  30  days,  2%  10  days  :  2  dozen  chisels  at  $3.20  ; 
5t2  dozen  10-inch  drawing  knives  at  $5.50 ;  1x^2  dozen 
ratchet  screw  drivers  at  $8.75,  subject  to  a  discount 
of  10%,  10%,  and  5%;  12  steel  shovels  at  87i^ ;  9 
spades  at  70^ ;  3  dozen  garden  rakes  at  $5 ;  4  dozen 
trowels  at  $1.35,  subject  to  a  discount  of  20%,  10%, 
and  10%;  4  dozen  cans  prepared  paint  at  $3.15;  480 
feet  of  f-inch  garden  hose  at  10^ ;  5  lawn  mowers  at 
$4.50,  subject  to  a  discount  of  16f%,  5%,  and  5%; 
8  dozen  boxes  of  4-inch  bolts  at  $2.35.  Make  out  bill 
showing  the  total  due  at  the  expiration  of  credit  and 
the  amount  due  on  the  cash  terms. 

9.  Use  current  date,  your  own  locality.  Naumkeag 
Company  bought  of  Childs  Provision  Company,  terms 
2%  for  cash  and  net  10  days :  150  barrels  Baldwin 
apples  at  $5.50 ;  25  barrels  greenings  at  $4.50 ;  45 
bushels  of  beans  at  $4.75 ;  90  bushels  of  potatoes  at 
$1.95,  subject  to  a  discount  of  10%  and  10%;  4  firkins 
of  butter,  65  pounds  each,  at  50^^ ;  5  firkins  creamery 
butter,  50  pounds  each,  at  48^ ;  6  boxes  American 
cheese,  50  pounds  each,  at  30^  ;  4  boxes  Young  America 
cheese,  33  pounds  each,  at  36^,  subject  to  a  discount  of 
10%.  5%,  and  2% ;  4  sacks  of  Rio  coffee,  140  pounds 
each,  at  48^ ;  2  bags  Java  coffee,  215  pounds  each,  at 
46^ ;  2  barrels  rice,  320  pounds  each,  at  16^ ;  1  barrel 
New  Orleans  molasses,  45  gallons,  at  35^;  1  barrel 
Porto  Rico  molasses,  45  gallons,  at  33^^,  subject  to  a 


RATIONAL  ARITHMETIC  57 

j 
discount  of  10%,  10%,  and  5%.     Make  out  bill  showing 

the  total  due  at  the  expiration  of  credit  and  the  amount 

due  on  the  cash  terms. 

10.  Use  current  date,  your  own  locality ;  seller, 
L.  A.  White,  Wholesale  Company ;  purchaser,  Parker 
Clothing  Company ;  terms,  net  60  days,  2%  10  days  : 
45  girls'  rain  capes  at  $4 ;  75  girls'  sweaters  at  $3.75  ; 
125  children's  rompers  at  $1.50 ;  75  boys'  rubber  coats 
at  $4.25,  subject  to  a  discount  of  15%,  10%,  and  2% ; 
35  voile  waists  at  $2.25  ;  40  organdie  waists  at  $3.50 ; 
70  pairs  women's  gloves  at  $1.60 ;  50  pairs  of  women's 
silk  gloves  at  $1.25;  14  dozen  women's  handkerchiefs 
at  $2 ;  12  dozen  men's  handkerchiefs  at  $2.25 ;  25 
men's  bath  robes  at  $5;  75  men's  shirts  at  $1.50; 
120  white  skirts  at  $3.24;  60  dress  skirts  at  $5.75; 
75  serge  dresses  at  $14.50;  50  ladies'  belts  at  75^, 
subject  to  a  discount  of  10%,  10%,  and  10%.  Make  out 
bill  showing  the  total  due  at  the  expiration  of  credit 
and  the  amount  due  on  the  cash  terms. 

11.  Use  current  date ;  your  own  locality ;  you  are 
the  seller ;  your  teacher  the  purchaser ;  terms,  net  30 
days,  2%  off  for  cash  :  3  #124  phonographs  at  $85 ; 
2  #062  phonographs  at  $125 ;  4  #68a  phonographs  at 
$75 ;  1  #300  phonograph  at  $250 ;  2  doz.  Operatic 
Records  at  $2  each ;  2  doz.  Standard  Song  Records 
at  $1.25  each;  2  doz.  Band  and  Orchestra  Records 
at  $1.50  each;  1  doz.  Comic  Monologue  Records  at 
S5i  each ;  1  doz.  Records  for  Dancing  at  S5^  each. 


COMMISSION 

Study  299  to  310  inclusive. 
References  311,  312. 

47.    Find  the  commission  and  net  proceeds : 


Sales 

Commission 

Charges 

1. 

$4342.50 

2i% 

$214.30 

2. 

$356.40 

3% 

$85 

3. 

$1256 

5% 

$13.25 

4. 

$8784.50 

34% 

$43.50 

5. 

$788.40 

4% 

$38.56 

Find  the  commission  and  gross  cost : 

Purchases  Commission  Charges 


6. 

$435.25 

3% 

$48.56 

7. 

$1840.50 

2i% 

$128.50 

8. 

$843.35 

3% 

$83.87 

9. 

$1258.40 

5% 

$324.50 

10. 

$5643.50 

8% 

$125.40 

References  311,  312. 

48.    1.    An  agent  sold  a  farm  for  $3690  at  2i%  com- 
mission.    What  was  his  commission  ? 

2.    I  sold   128  barrels  of  sugar,  each  weighing  350 
pounds,  at  9f  ^  a  pound.     What  was  my  commission  at 

n%  ? 

58 


RATIONAL  ARITHMETIC 


59 


3.  A  commission  merchant  sold  250  barrels  of  sugar, 
each  weighing  350  pounds,  at  9^^  a  pound  and  158 
barrels  of  molasses,  each  containing  48  gallons,  at  77^^ 
a  gallon.     Find  his  commission  at  2%. 

4.  A  real  estate  agent  sold  a  house  for  $4450  at  1^% 
commission.     What  sum  did  he  send  the  owner? 

5.  Rule  a  sheet  of  paper  and  copy  the  following 
account  sales,  making  the  necessary  extensions : 

ACCOUNT   SALES 

Albany,  N.  Y.,    August  14,   1919 
Sold  for  the  Account  of 

C.  F.  Adams,  Troy,  N.  Y. 
By  W.  H.  Smith,  Commission  Merchant. 


1919 
June 

July 


June 
July 


13 

28 
10 
15 


10 

8 

12 


100  bbl.  G.  M.  flour 

150  bbl.  G.  M.  flour 

200  bbl.  G.  M.  flour 

100  bbl.  G.  M.  flour 


Charges 


$10.50 
10.75 
10.25 
10.50 


Freight      $125 
Storage  15.50 

Guaranty      1% 


Cartage 


$27.00 
Insurance  5.20 

Commission      4% 


Net  Proceeds 


6.  Prepare  an  account  sales  under  date  of  January  10, 
for  5000  bushels  of  wheat,  sold  by  J.  J.  Campbell  &  Co., 
Springfield,  Mass.,  for  the  account  of  Walter  Bros., 
North  Adams,  Mass.,  Sales,  No.  25 ;  500  bushels  at 
$2.08,  December  30 ;  the  remainder  at  $2.  Charges  : 
freight,  $91 ;  cartage,  $15  ;  storage,  $15.50  ;  insurance, 
i%  ;   guaranty,  1%  ;   commission,  2i%. 


60  RATIONAL  ARITHMETIC 

7.  Prepare  an  account  sales  under  date  of  October  15 
for  the  account  of  J.  C.  Brown  &  Co.,  sold  by  Thomas 
Moody  &  Co.,  both  of  Chicago,  111.,  October  3 ;  50 
barrels  of  W.  R.  flour  at  $9.25  ;  100  barrels  of  K.  A. 
flour  at  $9.80.  Charges  :  freight,  $50  ;  cartage,  $14.50, 
both  under  date  of  Oct.  1 ;  Oct.  11,  storage,  150 
barrels  at  4^  a  barrel;     insurance,  i%;    commission, 

8.  Put  the  following  in  the  form  of  an  account  sales  : 
William  C.  Jones,  St.  Louis,  Mo.,  sold  for  account  of 
Charles  W.  Franklin,  Chicago,  111.,  the  following  goods 
August  4,  7  pieces  of  summer  silk,  284  yards,  at  $1.85 
August  13,  5  pieces  black  silk,  216  yards,  at   $1.50 
August  17,   16  pieces  calico,  798  yards,  at  19^;    Sep- 
tember 3,  19  pieces  alpaca,  548  yards,  at  38^;    Sep- 
tember   10,    25   pieces   diagonals,    587   yards,    at   75^. 
Charges :    August  1,  freight  and  cartage,  $65.48 ;    in- 
surance,   i% ;   commission,   4%.     Find    the    net    pro- 
ceeds. 

9.  Arrange  the  following  in  the  form  of  an  account 
sales :  E.  P.  Clark  &  Co.,  Peekskill,  N.  Y.,  sold  for 
account  of  John  Mason,  the  following :  November  1, 
300  bushels  potatoes  at  $1.95;  November  16,  200 
bushels  at  $1.85;  December  1,  240  bushels  at  $1.90. 
Charges :  November  1,  freight,  $85.45 ;  cartage,  2^^ 
per  bushel ;  storage  at  2^  per  bushel ;  commission,  4^%. 
Find  the  net  proceeds. 

10.  Rule  a  sheet  of  paper  and  copy  the  follow- 
ing account  purchase,  making  the  necessary  exten- 
sions. 


RATIONAL  ARITHMETIC 


61 


ACCOUNT   PURCHASE 

Utica,  N.  Y.,    May  10,  1919 
Purchased  by  F.  J.  Bowen  &  Co. 

For  the  Account  of  E.  L.  Green,     Rome,  N.  Y. 


1919 
April 

May 


24 
29 

7 
9 


3  half-ch.  J.  tea,  165  lb. 

4  half-ch.  O.  tea,  240  lb. 

5  half-ch.  J.  tea,  350  lb. 
8  mats  Rio  coffee,  600  lb. 


Charges 


Cartage 
Commission  5% 


$7.50 


Amount  charged  to  your  accoimt 


11.  In  accordance  with  the  foregoing  form  prepare 
an  account  purchase  of  tea  purchased  by  W.  L.  Thomas, 
Feb.  21,  for  the  account  of  Jones,  White  &  Co.,  both  of 
Boston,  Mass. ;  10  half-chests  of  J.  tea,  600  lb.,  at  ¥l(ji ; 
5  half-chests  O.  tea,  250  lb.,  at  45^;  5  cases  C.  tea, 
250  lb.,  at  50^ ;  8  half-chests  E.  B.  tea,  480  lb.,  at  49^2^. 
Charges  :   cartage,  $8.80  ;   commission,  4%. 

12.  Prepare  an  account  purchase  for  the  following : 
David  Carey,  New  York  City,  bought  of  the  account 
of  Henry  Grant  &  Co.  of  Newark,  N.  J.,  August  16, 
1916,  68  yd.  ^ancy  prints  at  25^;  42  yd.  colored  silk 
at  $1.25;  1  dozen  ladies'  felt  hats,  $30;  18  yd.  black 
cassimere  at  $1.50;  3  suits  boys'  clothing  at  $10. 
Charges :  packing  and  cartage,  $2.40.  Find  entire 
cost,  commission  being  5%. 


62  RATIONAL  ARITHMETIC 

13.  Prepare  an  account  purchase  for  the  following: 
March  18,  1916.  A.  B.  Morse  &  Co.,  Trenton,  N.  J., 
bought  for  account  of  Harris  &  Price,  of  Philadelphia, 
Pa.,  March  18,  150  bbl.  Dakota  flour  at  $9.75; 
80  bbl.  buckwheat  flour  at  $12.40 ;  480  bu.  ground 
feed  at  60^;  500  bu.  bran  at  30^;  20  bbl.  G.  M. 
flour  at  $11.  Cartage,  $7.90,  commission,  4%.  What 
is  the  entire  cost  ? 

14.  A  merchant  in  Boston  shipped  to  his  broker  in 
New  York  a  carload  of  potatoes,  967  bushels,  which 
were  sold  at  $2.25  a  bushel.  What  was  realized  on  the 
sale  if  the  broker  charged  4^%  for  selling  and  the 
freight  was  $67.96?  How  many  pounds  of  Java 
coffee  could  be  purchased  with  the  proceeds  of  the 
sale  of  potatoes  if  coffee  is  45^  a  pound  and  the  broker 
charges  2%  for  buying  ? 

References  313,  314,  315. 

49.  1.  A  commission  merchant  working  on  a  4% 
commission  earned  $240.50  for  selling  a  consignment 
of  flour.     What  did  the  flour  sell  for  ?  • 

2.  My  commission  for  selling  goods  at  2%  amounted 
to  $250.50.     What  was  the  selling  price  of  the  goods  ? 

3.  I  bought  goods,  receiving  $75.30  as  my  5%  com- 
mission. What  did  I  pay  for  the  goods  and  what  was 
the  total  cost  to  my  principal  ? 

4.  A  commission  merchant  whose  charge  is  1^% 
finds  that  his  total  receipts  for  commissions  for  3 
months  amount  to  $4856.  What  was  the  value  of 
his  sales  for  the  same  time.^ 


RATIONAL  ARITHMETIC  63 

5.  A  collector  charges  5%  for  his  services.  In  order 
that  he  may  clear  $3000  a  year,  what  must  his  collec- 
tions amount  to  ? 

6.  I  receive  $736.96  as  the  net  proceeds  of  the  sale 
of  goods  through  a  commission  merchant,  the  only 
charges  being  2%  for  commission.  What  was  the  gross 
sales  ? 

7.  My  lawyer  sends  me  $77.80  as  the  proceeds  of  a 
claim  which  he  has  collected  for  me.  What  was  the 
claim,  his  commission  being  5%  ? 

8.  We  have  received  a  check  for  $344.75  as  the  net 
proceeds  of  a  sale  on  which  the  commission  was  1^%. 
What  was  the  total  sales  and  what  was  the  commission  ? 

9.  I  sent  my  commission  merchant  $138.65  to  pay 
for  goods  purchased  by  him,  including  commission  of 
3i%.     What  did  he  pa}^  for  the  goods  ? 

10.  I  received  $185.40  from  J.  C.  Bryan  to  cover 
the  amount  of  goods  purchased  and  my  commission 
of  3%.     What  was  the  amount  of  the  purchase? 

11.  I  received  $564.20  with  instructions  to  purchase 
certain  supplies,  my  commission  to  be  li%.  What 
amount  will  I  invest  in  supplies  and  what  will  my  com- 
mission be  ? 

12.  A  commission  merchant  received  $1606  to  in- 
vest, after  deducting  a  commission  of  |%.  How  much 
can  he  invest  ? 

13.  We  received  $145.50  as  the  net  proceeds  of  a 
consignment.  The  rate  of  commission  was  2%  and 
other  charges  $2.50 ;  what  was  the  selling  price  of  the 
goods  ? 


64  RATIONAL  ARITHMETIC 

14.  A  commission  merchant  remitted  $704.29  after 
deducting  his  commission  of  lf%  and  other  charges 
amounting  to  $3.20.  What  was  the  selhng  price  of 
the  goods  and  what  was  the  commission? 

15.  My  commission  merchant  has  just  sent  me  an 
account  purchase  showing  gross  cost  to  be  $668.60. 
The  commission  w^as  figured  at  4% ;  the  other  charges 
amounted  to  $8.20.  What  was  the  prime  cost  of  the 
purchase  ? 

16.  I  paid  a  real  estate  broker  $225  for  selling  a 
house  and  lot.  This  sum  included  his  commission  of 
5%  and  other  expenses  amounting  to  $50.  What  sum 
did  the  house  sell  for  ? 

17.  A  commission  merchant  received  $12,500  to 
invest  in  cotton  after  deducting  his  commission  of  5%. 
What  sum  does  he  invest  .^^  How  many  bales  of  400 
pounds  can  be  bought  at  35^  a  pound? 

18.  I  shipped  my  agent  in  New  York  950  tons  of  hay 
which  he  sold  for  me  at  $22.50  a  ton.  Charges  were  :  for 
freight,  $950;  cartage,  $327.50;  storage,  $.85  a  ton; 
his  commission,  2^%.  I  instructed  him  to  invest  the  pro- 
ceeds in  wheat  for  me.  If  he  charged  me  at  the  same 
rate  for  investing  that  he  did  for  selling  the  hay,  how 
much  did  he  invest  and  what  was  his  entire  commission  ? 

19.  I  bought  a  lot  of  apples  for  $6.50  a  barrel  on 
3%  commission.  If  my  commission  amounted  to 
$81.51,  how  many  barrels  did  I  buy? 

20.  I  received  $893.03  with  instructions  to  invest  it 
in  apples  after  deducting  a  commission  of  5%.  How^ 
many  barrels  can  I  buy  at  $5.50  a  barrel? 


RATIONAL  ARITHMETIC  65 

Reference  316. 
50.     What  is  the  rate  of  commission  : 

1.  If  the  prime  cost  of  merchandise  is  $480  and  the 
commission  for  buying  is  $4.20  ? 

2.  If  the  net  proceeds  is  $944.40  and  the  commission 
is  $15.60? 

3.  If  the  first  cost  is  $3264  and  the  commission  for 
buying  is  $4.08  ? 

4.  If  the  gross  proceeds  is  $3200  and  the  commission 
for  selhng  is  $128? 

5.  If  a  commission  merchant  receives  $37.02  for 
selHng  $1234  worth  of  goods  ? 

6.  If  a  commission  merchant  receives  $85.75  for 
selling  $3430  worth  of  goods  ? 

7.  A  real  estate  dealer  bought  a  house  for  a  client, 
paying  $8750  for  it.  His  charges  were  $262.50.  What 
was  his  rate  of  commission  ? 

8.  A  commission  merchant  bought  346  barrels  of 
apples  at  $5.75,  receiving  $65.74  for  his  commission ; 
other  charges  amounted  to  $43.13.  What  is  the  gross 
purchase  and  what  is  his  rate  of  commission  ? 

9.  A  lawyer  earned  $37.40  for  collecting  a  claim  on 
5%  commission.     What  was  the  claim  ? 

10.  A  merchant  sent  his  principal  $325.23  as  the 
net  proceeds  of  a  consignment  which  he  sold  for  $343.43. 
What  was  the  rate  of  commission  ? 


66  RATIONAL  ARITHMETIC 

GENERAL   PROBLEMS   IN    COMMISSION 
References  311  to  316  inclusive. 

51.  1.  My  agent  sells  $1400  worth  of  goods  for  me 
at  3%  commission.     What  amount  must  he  remit  .^^ 

2.  An  agent  sells  goods  for  $2468.  His  charges  are  : 
commission,  2^% ;  storage,  insurance,  etc.,  $125. 
What  are  the  net  proceeds? 

3.  I  sold  for  a  Chicago  firm  as  follows :  950  bu. 
corn  at  85^  a  bushel;  120  bbl.  of  pork  at  $15.50  per 
barrel.  My  commission  on  the  corn  was  1^^  a  bushel 
and  the  commission  on  the  pork  was  2^%  .  Charges 
were :  freight,  $134 ;  storage,  $67 ;  advertising,  $63. 
What  were  the  net  proceeds  due  my  employer  ? 

4.  I  purchased  for  a  South  American  firm  goods 
valued  at  $3750.  My  commission  was  to  be  2%.  For 
what  sum  must  I  draw.^^ 

5.  I  have  just  received  $4168  as  the  net  proceeds 
of  a  consignment.  The  figures  in  the  Account  Sales 
are  blurred  and  I  am  unable  to  read  either  the  amount 
of  the  sales  or  the  rate  of  commission,  which  is  $145.84. 
Ascertain  both. 

6.  I  paid  a  real  estate  dealer  $215  for  selling  a  house 
and  lot  on  5%  commission.  Advertising  and  other 
expenses  amounted  to  $50.  What  amount  does  the 
sale  net  me  ? 

7.  I  have  purchased  for  a  Southern  firm  45,620  feet 
of  pine  lumber  at  $16.25  a  thousand ;  34,257  feet  of 
hemlock  boards  at  $15  a  thousand ;  37,250  feet 
of  spruce  at  $18.60  a  thousand.     My  commission  is 


RATIONAL  ARITHMETIC  67 

Si%  and  my  other  charges  amounted  to  $214.  For 
what  sum  shall  I  draw  on  the  firm  to  cover  the  cost  of 
my  purchase  and  charges  ? 

8.  A  commission  merchant  received  $648.40  to  in- 
vest in  wool  after  deducting  all  his  expenses.  How 
much  did  he  pay  for  the  wool  if  his  commission  for 
buying  was  2%  and  his  other  charges  amounted  to 
$13.50? 

9.  A  commission  merchant  received  a  consignment 
of  Q'i5  barrels  of  flour  on  which  he  paid  $84  for  freight ; 
$12.75  for  cooperage ;  $22.50  for  storage ;  $19.50  for 
cartage.  He  sold  110  barrels  at  $9.25  a  barrel;  175 
barrels  at  $10.75;  123  barrels  at  $11.25,  and  the  re- 
mainder at  $10.  His  commission  for  selling  was  2%. 
What  was  the  net  proceeds  ? 

10.  The  gross  cost  of  goods  purchased  through  an 
agent  was  $1221.  If  the  commission  was  $6  and  the 
other  charges  $15,  what  was  the  rate  of  commission.'^ 

11.  I  have  received  $325  to  invest  in  apples  after 
deducting  all  expenses.  How  many  barrels  could  I 
buy  at  $5.75  a  barrel  ;  charges  being  3%  for  commis- 
sion ;  2^^  a  barrel  for  dray  age ;  12^  a  barrel  for  freight ; 
$5  for  advertising  ?  What  was  the  unexpended  balance, 
if  any  ? 

12.  I  sent  $1547  to  my  agent  in  Chicago  with  in- 
structions to  buv  wheat  after  deducting  his  commission 
of  3%.     How  much  did  he  invest  in  wheat? 

13.  A  commission  merchant  received  from  a  specu- 
lator $2091  to  invest  in  corn  after  deducting  his  com- 
mission of  2^%.     He  was  instructed  to  hold  the  corn 


68  RATIONAL  ARITHMETIC 

subject  to  the  purchaser's  order.  After  an  advance 
in  value  he  was  ordered  to  sell,  and  did  so,  obtaining 
$1.50  a  bushel.  After  deducting  his  commission  of 
2|%  and  $20  for  storage,  he  paid  the  speculator  $2220 
as  the  balance  due  him.  What  did  the  commission 
agent  pay  a  bushel  for  the  corn  ? 

14.  A  commission  merchant's  regular  charges  were 
3%  for  selling  and  2%  for  guaranteeing  the  purchase 
price.  If  he  remitted  $6842  to  his  principal  as  the  net 
proceeds  of  the  sale,  what  did  the  goods  sell  for.^ 

15.  A  commission  merchant  sold  625  barrels  of 
potatoes  at  $11.25  a  barrel  and  invested  the  proceeds 
in  wheat  at  85  j^  a  bushel,  first  deducting  a  commission 
of  2%  for  buying  and  2%  for  selling.  How  many 
bushels  of  wheat  could  he  buy  and  what  was  the  un- 
expended balance,  if  any  .^ 

Find  the  missing  quantities  in  the  following : 

Sale  Commission  Rate  of  Com.  Net  Proceeds 

16.  1246.50 5%      

17.  234.40      2i%      

18.  5450.       272.50      


19. 47.49      1535.51 

20.  6254.50      6004.32 

21. 4%      3317.76 

22.  8564.50 

23.  1327.  79.62 


0 


24. 33.87       6%      

25.   856.40 834.99 


TIME 

Study  carefully  317  to  325  inclusive. 
Reference  238. 

52.    By  compound  subtraction  find  the  time  from : 

1.  Jan.  3,  1910  to  March  5,  1916. 

2.  Dec.  28,  1912  to  June  11,  1914. 

3.  May  13,  1909  to  Dec.  2,  1911. 

4.  June  28,  1910  to  May  12,  1911. 

5.  April  3,  1912  to  Oct.  14,  1913. 

6.  Feb.  28,  1912  to  Jan.  1,  1914. 

7.  July  4,  1914  to  April  14,  1916. 

8.  May  13,  1905  to  Dec.  9,  1915. 

9.  March  9,  1908  to  Feb.  6,  1912. 

10.  June  11,  1909  to  April  3,  1916. 

11.  Dec.  6,  1914  to  Aug.  7,  1917. 

12.  Nov.  28,  1913  to  June  28,  1917. 

13.  Sept.  8,  1907  to  Aug.  21,  1915. 

14.  May  30,  1905  to  July  20,  1906. 

15.  Oct.  9,  1909  to  June  11,  1916. 

16.  June  17,  1913  to  May  16,  1916. 

17.  Feb.  6,  1908  to  May  9,  1917. 

18.  July  8,  1911  to  Aug.  2,  1912. 

19.  April  4,  1910  to  March  17,  1913. 

69 


70  RATIONAL  ARITHMETIC 

20.  July  4,  1908  to  May  22,  1916. 

21.  Dec.  13,  1909  to  Nov.  12,  1917. 

22.  Feb.  3,  1907  to  May  11,  1915. 

23.  June  18,  1911  to  Aug.  14,  1917. 

24.  April  21,  1916  to  Dec.  16,  1917. 

25.  Aug.  17,  1909  to  June  11,  1916. 

26.  Jan.  30,  1912  to  Feb.  18,  1912. 

27.  Oct.  27,  1908  to  July  14,  1915. 

28.  Nov.  13,  1905  to  Jan.  26,  1909. 

29.  April  5,  1910  to  Oct.  9,  1917. 

30.  March  3,  1906  to  June  8,  1916. 

31.  May  30,  1913  to  March  12,  1914. 

32.  Sept.  24,  1906  to  April  30,  1915. 

33.  Nov.  27,  1909  to  Sept.  28,  1917. 

34.  Sept.  13,  1911  to  May  11,  1914. 

35.  June  20,  1905  to  Oct.  14,  1916. 

36.  Dec.  11,  1912  to  Feb.  4,  1917. 

37.  Nov.  13,  1914  to  Aug.  24,  1917. 

38.  May  11,  1908  to  July  31,  1912. 

39.  Feb.  28,  1906  to  June  30,  1913. 

40.  Nov.  8,  1913  to  Sept.  3,  1917. 

41.  Jan.  1,  1908  to  July  11,  1915. 

42.  Aug.  3,  1913  to  June  11,  1916. 

43.  Nov.  21,  1908  to  Oct.  9,  1917. 

44.  March  3,  1911  to  June  16,  1912. 

45.  April  13,  1910  to  Feb.  22,  1916. 

46.  Dec.  31,  1909  to  Feb.  22,  1916. 

47.  Aug.  19,  1912  to  April  3,  1917. 


RATIONAL  ARITHMETIC  71 

48.  Oct.  4,  1913  to  June  8,  1915. 

49.  June  11,  1914  to  May  3,  1917. 

50.  Sept.  3,  1909  to  June  21,  1916. 

Reference  239. 

53.  Find  the  time  in  exact  days  from  : 

1.  Jan.  13,  1908  to  Nov.  12,  1908. 

2.  April  6,  1915  to  June  29,  1915. 

3.  Dec.  14,  1908  to  Mav  12,  1909. 

4.  May  8,  1916  to  Nov.  30,  1916. 

5.  June  11,  1915  to  Oct.  27,  1915. 

6.  Aug.  12,  1916  to  July  3,  1917. 

7.  Nov.  16,  1915  to  Jan.  18,  1916. 

8.  Aug.  17,  1914  to  Dec.  31,  1914. 

9.  July  30,  1913  to  Jan.  1,  1914. 

10.  May  27,  1906  to  April  28,  1907. 

11.  Feb.  12,  1908  to  Nov.  13,  1908. 

12.  Dec.  6,  1916  to  Aug.  6,  1917. 

13.  May  12,  1916  to  Oct.  28,  1916. 

14.  June  11,  1909  to  Jan.  3,  1910. 

15.  Oct.  24,  1912  to  May  16,  1913. 

16.  July  16,  1911  to  Dec.  25,  1911. 

17.  Sept.  3,  1908  to  May  27,  1909. 

18.  Aug.  18,  1915  to  Nov.  26,  1915. 

19.  Sept.  27,  1908  to  May  13,  1909. 

20.  Feb.  13,  1914  to  Dec.  6,  1915. 

21.  June  17,  1915  to  April  8,  1916. 

22.  July  4,  1916  to  Dec.  25,  1916. 


72  RATIONAL  ARITHMETIC 

23.  May  2,  1914  to  April  18,  1915. 

24.  Dec.  16,  1913  to  Sept.  29,  1914. 

25.  May  28,  1916  to  Jan.  2,  1917. 

26.  Sept.  21,  1914  to  Dec.  1,  1914. 

27.  April  13,  1915  to  March  30,  1916. 

28.  Oct.  27,  1914  to  June  3,  1915. 

29.  Nov.  28,  1906  to  Dec.  13,  1906. 

30.  Jan.  31,  1914  to  Jan.  3,  1915. 

31.  May  6,  1911  to  April  14,  1912. 

32.  Dec.  25,  1915  to  July  4,  1916. 

33.  June  22,  1909  to  May  23,  1910. 

34.  April  28,  1916  to  March  17,  1917. 

35.  Sept.  8,  1916  to  March  17,  1917. 

36.  Dec.  4,  1913  to  Feb.  12,  1914. 

37.  Dec.  25,  1916  to  July  4,  1917. 

38.  Aug.  31,  1914  to  May  12,  1915. 

39.  Dec.  30,  1915  to  Sept.  9,  1916. 

40.  June  11,  1914  to  Jan.  1,  1915. 

41.  April  3,  1915  to  Jan.  4,  1916. 

42.  Dec.  28,  1914  to  May  19,  1915. 

43.  Jan.  7,  1915  to  Sept.  8,  1915. 

44.  Feb.  28,  1916  to  Jan.  12,  1917. 

45.  June  3,  1912  to  May  21,  1913. 

46.  May  14,  1911  to  Nov.  28,  1911. 

47.  Sept.  9,  1912  to  June  11,  1913. 

48.  Oct.  28,  1916  to  July  14,  1917. 

49.  May  28,  1914  to  April  7,  1915. 

50.  Dec.  8,  1915  to  Sept.  29,  1916. 


INTEREST 

Study  carefully  317  to  352  inclusive. 
References  333,  334. 

54.    Find  the  interest  on  : 

1.  $462.40  for  2  mo.  6  da.  at  6%. 

2.  $385.60  for  3  mo.  18  da.  at  6%. 

3.  $460.75  for  1  mo.  8  da.  at  6%. 

4.  $200  for  1  mo.  3  da.  at  6%. 

5.  $260.70  for  72  da.  at  6%. 

6.  $650  for  4  mo.  9  da.  at  6%. 

7.  $450  for  6  mo.  15  da.  at  6%. 

8.  $124  for  8  mo.  14  da.  at  6%. 

9.  $285.50  for  93  da.  at  6%. 

10.  $450.65  for  1  yr.  3  mo.  13  da.  at  4%. 

11.  $562.30  for  5  mo.  13  da.  at  4%. 

12.  $287.95  for  9  mo.  16  da.  at  8%. 

13.  $396.40  for  1  yr.  3  mo.  14  da.  at  8%. 

14.  $756  for  8  mo.  18  da.  at  5%. 

15.  $468.40  for  2  yr.  5  mo.  6  da.  at  7%. 

16.  $400  for  29  da.  at  7i%. 

17.  $216.80  for  1  yr.  5  mo.  6  da.  at  4^%. 

18.  $375.90  for  7  mo.  28  da.  at  4^%. 

19.  $219.76  for  1  yr.  3  mo.  18  da.  at  9%. 

73 


74  RATIONAL  ARITHMETIC 

20.  $468.75  for  10  mo.  13  da.  at  3%. 

21.  $240  for  3  mo.  22  da.  at  12%. 

22.  $375.80  for  1  mo.  10  da.  at  4%. 

23.  $265.72  for  3  mo.  7  da.  at  4i%. 

24.  $486.85  for  4  mo.  3  da.  at  6%. 

25.  $265.72  for  1  yr.  7  mo.  14  da.  at  8%. 

26.  $380.60  for  8  mo.  15  da.  at  7%. 

27.  $450  for  3  mo.  17  da.  at  7i%. 

28.  $264.40  for  5  mo.  11  da.  at  9%. 

29.  $942.60  for  2  yr.  9  mo.  16  da.  at  5%. 

30.  $384.32  for  3  mo.  12  da.  at  4^%. 

31.  $295.73  for  1  yr.  6  mo.  15  da.  at  3%. 

32.  $389.60  for  11  mo.  18  da.  at  6%. 

33.  $560.35  for  2  mo.  14  da.  at  8%. 

34.  $387.60  for  6  mo.  5  da.  at  4%. 

35.  $418.62  for  3  mo.  28  da.  at  4i%. 

36.  $560.32  for  7  mo.  19  da.  at  6%. 

37.  $362.40  for  1  yr.  8  mo.  20  da.  at  4>^%. 

38.  $271.35  for  4  mo.  24  da.  at  7%. 

39.  $361.75  for  44  da.  at  7%. 

40.  $285.60  for  2  yr.  8  mo.  24  da.  at  7i%. 

41.  $397.80  for  7  mo.  12  da.  at  5%. 

42.  $184.25  for  8  mo.  21  da.  at  4^%. 

43.  $1495.60  for  1  yr.  2  mo.  13  da.  at  6%. 

44.  $372.75  for  11  mo.  8  da.  at  7%. 

45.  $175.43  for  2  yr.  7  mo.  14  da.  at  7i%. 

46.  $295.60  for  9  mo.  24  da.  at  6%. 

47.  $362.70  for  8  mo.  21  da.  at  6%. 


RATIONAL   ARITHMETIC  75 

48.  $467.80  for  1  yr.  3  mo.  5  da.  at  3%. 

49.  $284.60  for  9  mo.  13  da.  at  8%. 

50.  $575.80  for  6  mo.  18  da.  at  4^%. 

References  335,  336. 
55.    Find  the  interest  on  : 

1.  $600  from  Jmie  1,  1916  to  Aug.  13,  1916  at  6%. 

2.  $360  from  Oct.  3,  1914  to  June  3,  1915  at  5%. 

3.  $180  from  Dec.  6,  1915  to  July  13,  1916  at  8%. 

4.  $840.75  from  May  12,  1912  to  Dec.  8,  1914  at  7%. 

5.  $454.54  from  Jan.  16,  1916  to  Oct.  28,  1916  at  5%. 

6.  $544.44  from  April  5,  1916  to  Jan.  1,  1917  at  1\%, 

7.  $850  from  July  18,  1914  to  Dec.  31,  1916  at  9%. 

8.  $809  from  Sept.  13,  1912  to  June  11,  1915  at  5%. 

9.  $256  from  Nov.  28, 1913  to  Mar.  12, 1914  at  1\%. 

10.  $660.80  from  Aug.  17,  1914  to  Jan.  3,  1916  at  7%. 

11.  $840  from  April  1,  1914  to  Jan.  3,  1916  at  7%. 

12.  $629  from  Nov.  13,  1908  to  Aug.  4,  1910  at  \\%, 

13.  $548  from  Jan.  30,  1912  to  Sept.  28,  1914  at  7%. 

14.  $465.10  from  Oct.  2,  1913  to  Sept.  12,  1914  at  8%. 

15.  $654  from  Feb.  12,  1914  to  Dec.  21,  1916  at  6%. 

16.  $360  from  June  16,  1909  to  Sept.  30,  1914  at  4^%. 

17.  $126  from  Aug.  28,  1907  to  Feb.  16,  1912  at  4%. 

18.  $480  from  April  6,  1910  to  June  30,  1914  at  5%. 

19.  $1000  from  Oct.  13,  1914  to  Nov.  28,  1915  at  6%. 

20.  $975  from  May  12,  1916  to  Nov.  11,  1916  at  9%. 

21.  $649.24  from  Jan.  1,  1914  to  April  3,  1916  at  6%. 

22.  $100  from  Oct.  2,  1915  to  Feb.  12,  1916  at  5%. 


76  RATIONAL  ARITHMETIC 

23.  $654  from  Dec.  12,  1915  to  Aug.  7,  1916  at  5%. 

24.  $962  from  March  8,  1915  to  Aug.  7,  1916  at  4^%. 

25.  $269.05  from  Dec.  3,  1909  to  Sept.  8,  1915  at  4%. 

26.  $680  from  April  21,  1914  to  Nov.  2,  1914  at  6%. 

27.  $500  from  Aug.  19,  1913  to  May  28,  1915  at  8%. 

28.  $85  from  Nov.  12,  1914  to  April  3,  1915  at  4%. 

29.  $450  from  Feb.  9,  1916  to  Dec.  21,  1916  at  5%. 

30.  $240  from  Jan.  8,  1913  to  June  9,  1915  at  7%. 

31.  $400  from  Sept.  12,  1909  to  Oct.  9,  1914  at  7^%. 

32.  $560  from  July  4,  1910  to  Sept.  8,  1914  at  4i%. 

33.  $200  from  March  2,  1913  to  Sept.  27,  1913  at  9%. 

34.  $460  from  Dec.  8,  1916  to  Feb.  3,  1917  at  3%. 

35.  $296.50  from  May  8,  1904  to  June  11,  1913  at  8%. 

36.  $320.60  from  Oct.  28,  1913  to  July  7,  1915  at  4%. 

37.  $576  from  Jan.  23,  1909  to  Aug.  13,  1912  at  6%. 

38.  $320.60  from  Dec.  6, 1914  to  June  28, 1916  at  4>i%. 

39.  $720.14  from  July  11, 1912  to  Aug.  29, 1916  at  8%. 

40.  $365.40  from  Sept.  14, 1911  to  Feb.  18, 1915  at  7%. 

41.  $428.60  from  May  11, 1916  to  Nov.  8, 1916  at  12%. 

42.  $576.80fromJune20,1909toDec.31,1909atl0%. 

43.  $162.38  from  Oct.  27,  1914  to  Dec.  3,  1916  at  8%. 

44.  $316.20  from  July  26,  1913  to  Oct.  19,  1914  at  5%. 

45.  $483.90  from  Jan.  8,  1915  to  July  4,  1916  at  4i%. 

46.  $265.70  from  Sept.  4,  1915  to  May  30,  1916  at  9%. 

47.  $456.75  from  Dec.  9, 1914  to  June  30, 1916  at  U%. 

48.  $195  from  Nov.  19,  1913  to  April  19,  1914  at  4%. 

49.  $362.80  from  Sept.  12,  1914  to  Aug.  8,  1915  at  6%. 

50.  $195.64  from  Aug.  14,  1915  to  May  3,  1916  at  3%. 


RATIONAL  ARITHMETIC  77 

ACCURATE   INTEREST 
References  S'2G,  327,  328 ;  337  to  343  inclusive. 

While  both  methods  are  used  in  business,  the  one 
explained  in  340  is  better  because  of  the  infrequency 
with  which  accurate  interest  is  used. 

56.  Find  the  accurate  interest  of  : 

1.  $1436  for  '295  days  at  6% ;   at  8%. 

2.  $484.50  for  193  days  at  6% ;   at  7i%. 

3.  $956.35  for  1  year  214  days  at  6% ;   at  4^%. 

4.  $632  for  462  days  at  6% ;   at  4^%. 

5.  $1284.50  from  Aug.  8, 1916  to  Jan.  12, 1917  at  8%. 

6.  $543.32  from  Apr.  7,  1916  to  Feb.  25,  1917  at  5%. 

7.  $246.50  from  Jan.  5, 1915  to  May  27, 1916  at  4i%. 

8.  $3432.40  from  June  13,  1914  to  Feb.  29,  1916  at 
Si%. 

9.  £  120  9s  Sd  for  214  days  at  8%. 

10.  £253  ll5  10^  for  313  days  at  5%. 

11.  £586  Us  4>d  for  1  year  246  days  at  4i%. 

12.  £732  Ids  9d  for  2  years  97  days  at  3%. 

Note  :  Change  English  money  to  pounds ;  see  232,  and  then 
apply  343.     Reduce  resulting  decimal  to  lower  denomination,  231. 

TO   FIND   TIME 
References  346,  347,  348. 

57.  1.  In  what  time  will  $417.40  produce  $7.43  at 
3i%  interest  ? 

2.    How  long  will  it  take  $325.80  on  interest  at  10% 
to  produce  $30.86  ? 


78  RATIONAL   ARITHMETIC 

3.  $895.80  earned  $16.50  at  7%  interest.  How  long 
was  the  money  at  interest  ? 

4.  How  long  will  it  take  $9500  to  earn  $1524.75  at 

9%? 

5.  I  loaned  $592.25  to  my  brother  at  5%  interest. 
He  paid  me  $13.74  interest.  How  long  did  he  have  the 
money  ? 

6.  $182.40  drawing  interest  at  4%  earned  $9.48. 
How  long  was  it  invested  ? 

7.  $4150.30  was  invested  at  8%  long  enough  to  earn 
$295.58.     How  long  was  it  invested  ? 

8.  $318.60  at  3%  would  require  how  long  to  produce 
$7.38  interest.^ 

9.  I  loaned  $7500  at  7%.  When  the  loan  was  paid 
I  received  $7702.71.  For  what  time  was  the  loan 
made  ? 

10.  I  received  a  check  for  $533.85  to  cancel  a  loan  of 
$519.75  effected  at  4^%.  How  long  had  the  loan  been 
standing  ? 

TO   FIND   RATE 
References  349,  350. 

58.  1.  At  what  rate  will  $1836  earn  $23,46  in  115 
days  ? 

2.  The  interest  on  $852  for  18  days  is  $2.13.  What 
is  the  rate  ? 

3.  In  93  days  $75  increases  at  interest  $1.55.  What 
is  the  rate  ^ 


RATIONAL  ARITHMETIC  79 

4.  In  144  days  $375  amounts  to  $381.75.     What  is 
the  rate  ? 

5.  At  what  rate  would  $982  be  placed  at  interest 
for  2  mo.  12  da.  to  earn  $9.82  ? 

6.  A   loaned   $2160   for    5    yr.    9    mo.    1    da.      It 
amounted  to  $3091.95.     What  rate  did  he  charge? 

7.  $588  on  interest  for  2  yr.  3  mo.  and  18  da.  earns 
$67.62.     What  is  the  rate  ? 

8.  At  what  rate  must  $3500  be  placed  at  interest 
for  4  mo.  and  15  da.  to  amount  to  $3605  ? 

9.  $296  produced  $3.70  in  45  days.     At  what  rate 
was  it  invested  ? 

10.  $810  amounts  to  $829.44  in  6  months  and  12 
days.     What  was  the  rate  ? 

11.  $750  on  interest  from  March  1,  1916  to  August  7, 
1916  earns  $26.50.     What  is  the  rate.^ 

12.  From  April  7,   1915   to  October  2,   1915,  $360 
earns  $16.02.     What  is  the  rate  ? 

TO   FIND   PRINCIPAL 
References  351  to  356. 

59.     1.    What    principal    will    be    required    to    earn 
$13.18  in  11  mo.  11  da.  .^ 

2.  W'hat  principal  will  be  required  to  earn  $39.92 
in  192  days  at  3^%  ? 

3.  How  much  money  invested  at  5^%  for  86  days 
will  earn  $24.62? 


80  RATIONAL   ARITHMETIC 

4.  The  interest  is  $16,  time  5  months  and  18  days, 
rate  8%.     What  is  the  principal  ? 

5.  At  9%  the  interest  for  3  mo.  27  da.  is  $150.07. 
What  is  the  principal  ? 

6.  W^hat  principal  will  in  6  mo.  13  da.  at  6%  amount 
to  $720.55 .? 

7.  What  principal  will  in  8  mo.  25  da.  at  5%  earn 
$15.73? 

8.  In  4  mo.  9  da.  at  4%  what  principal  will  amount 
to  $591.61.^ 

9.  At  6%  what  principal  will  in  1  yr.  9  mo.  28  da. 
yield  $74.99? 

10.  Money  invested  for  1  yr.  7  mo.  14  da.  at  4^% 
amounts  to  $531.30.     What  was  the  principal? 

11.  A  certain  sum  of  money  on  interest  at  3%  from 
May  19,  1910  to  July  15,  1915  earns  $392.35.  What  is 
the  investment  ? 

12.  W^hat  sum  was  loaned  on  April  5,  1910  at  7%  if 
it  were  paid  by  check  for  $942.66  on  Jan.  9,  1916? 

13.  What  principal  on  interest  from  October  25, 
1905  to  May  21,  1912  at  12%  would  amount  to 
$566.44  ? 

14.  What  sum  on  interest  from  July  15,  1915  to 
October  19,  1915  at  7%  will  earn  $14.90? 

15.  What  sum  on  interest  from  May  13,  1910  to 
August  25,  1919  would  amount  to  $11,446.47? 


RATIONAL  ARITHMETIC  81 

GENERAL   PROBLEMS   IN   INTEREST 

Before  attempting  to  solve  the  following  problems,  the  student 
should  be  thoroughly  familiar  with  the  entire  subject  of  interest  as 
presented  in  paragraphs  317  to  356  inclusive.  The  following  prob- 
lems comprise  a  series  of  tests  on  the  subject  of  interest  and  are  not 
graded  according  to  difficulty,  but  are  arranged  as  such  problems 
might  present  themselves  in  business.  Solve  each  in  the  simplest 
possible  way. 

60.  1.  A  note  for  $852  dated  May  2,  1915  with 
interest  at  5%  was  paid  on  Feb.  14,  1916  by  certified 
check.  What  was  the  amount  of  the  check,  time 
being  computed  in  exact  days  ? 

2.  I  have  just  received  a  legacy  of  $5000.  I  have 
an  obligation  of  $395  which  will  be  due  in  1  yr.  6  mo. 
and  14  da.  from  to-day.  I  have  decided  to  set  aside 
enough  of  my  legacy  at  4%  interest  to  pay  the  obliga- 
tion when  it  is  due.     How  much  will  I  have  to  set  aside  ? 

3.  Brown  borrowed  of  me  $400  at  7%,  Jones  $647  at 
5%,  and  Smith  $398  at  4^%.  Brown's  loan  ran  1  year 
8  months  and  11  days,  Jones'  ran  6  months  and  19  days, 
and  Smith's  ran  297  days.  What  were  my  total  re- 
ceipts for  interest  ? 

4.  After  a  loss  by  a  fire  the  insurance  company  has 
agreed  to  pay  me  $4340  in  full  settlement  of  the  claim. 
They  will  pay  this  amount  in  full  at  the  end  of  60  days 
or  will  make  a  cash  settlement,  deducting  2%.  Which 
proposition  should  I  accept  and  what  will  I  gain  by  so 
doing,  money  being  worth  6%  ? 

5.  On  May  26,  1915  I  gave  my  note  for  $1000  for 
6  months  at  5%.     November  26,  1915  I  paid  the  note 


82  RATIONAL  ARITHMETIC 

and   accumulated   interest,   reckoned    on   the  basis  of 
exact  days.     How  much  did  I  pay? 

6.  A  bill  of  $780.14  due  on  March  3,  1909,  was  not 
paid  until  October  15,  1909,  when  it  w^as  settled  with 
interest  at  6%.  What  was  the  amount  paid,  computing 
time  in  exact  days  ? 

7.  A  house  costing  $7500  rents  for  $60  a  month. 
What  rate  of  interest  does  the  investment  pay  if  the 
annual  expenses,  including  repairs,  taxes,  etc.,  amount 

to  $250  ? 

8.  What  will  be  the  difference  in  the  amount  of 
interest  involved  on  a  claim  for  $85,  running  from 
May  12,  1909  to  October  19,  1909  at  8%,  between 
the  amount  due  by  computing  the  time  in  exact  da^^s 
and  computing  the  time  in  months  and  da^^s  ? 

9.  On  May  1,  1915,  I  bought  a  bill  of  hides  at 
$15,300,  on  60  days'  credit,  2%  off  for  cash,  and  bor- 
rowed the  money  at  6%  on  the  hides  as  security  to 
accept  the  cash  price,  giving  my  note  for  60  days.  At 
the  expiration  of  the  note  I  had  tanned  the  hides  at 
an  expense  of  $1500  and  sold  them  at  an  advance  of 
25%  on  the  price  paid  for  them.  After  paying  my 
note,  what  was  my  net  profit  ? 

10.  I  bought  a  bill  of  $1450  subject  to  a  discount  of 
20%,  10%,  and  5%,  with  an  additional  discount  of  ^Z% 
for  cash,  and  borrowed  the  money  to  pay  for  it  at  5%. 
After  45  days  I  sold  the  goods  at  the  same  list  price, 
subject  to  a  discount  of  25%,  2%  extra  for  cash,  re- 
ceiving cash  settlement,  and  paid  my  loan.  What  was 
my  profit  ? 


PARTIAL  PAYMENTS 

Study  carefully  357  to  370  inclusive,  before  attempting  to  work 
on  partial  payments. 

References  371,  372,  373. 
Use  the  United  States  Rule. 

61.  1.  What  is  the  balance  due  on  July  1,  1916  on 
a  note  for  $1500  dated  February  1,  1913,  upon  which 
the  following  payments  were  made :  July  24,  1913, 
$250;  Aug.  7,  1913,  $100;  March  9,  1915,  $50;  Jan. 
1,  1916,  $300;    interest  at  the  rate  of  5%? 

2.  What  is  the  balance  due  on  December  31,  1917 
on  a  note  for  $1400  dated  May  2,  1915,  bearing  in- 
terest at  6%  and  having  the  following  indorsements : 
July  1,  1915,  $200;  September  25,  1915,  $90;  Febru- 
ary 28,  1916,  $175;    May  19,  1917,  $475? 

3.  On  April  21,  1913,  I  gave  my  note  for  $3550, 
payable  in  three  years,  interest  at  4 5%.  I  paid  as 
follows:  Oct.  15,  1913,  $125;  March  3,  1914,  $125; 
July  20,  1914,  $875.  How  much  will  be  required  to 
settle  the  note  at  maturitv  ? 

4.  W.  A.  Jones  gave  a  note  on  June  4,  1914  to  Frank 
Brown  for  $1285.50  with  interest  at  7%.  He  made 
payments  as  follows:  Dec.  15,  1914,  $340;  Feb.  17, 
1915,  $330;  March  11,  1915,  $400.  What  was  due  on 
Mav  15,  1915  ? 

83 


84  RATIONAL  ARITHMETIC 

5.  Find  the  value  of  a  note  for  $2500,  given  Oct.  10, 
1915,  with  interest  at  4^%,  and  on  which  the  following 
payments  have  been  indorsed :  Jan.  5,  1916,  $815 ; 
June  18,  1916,  $350;  Oct.  10,  1916,  $250;  Jan.  17, 
1917,  $150.     Settlement  was  made  Sept.  6,  1917. 

6.  On  a  note  of  $3080,  dated  Oct.  1,  1915,  the  follow- 
ing payments  have  been  made,  interest  being  at  6%  : 
Dec.  31,  1915,  $300;  Feb.  29,  1916,  $50;  June  5,  1916, 
$500  ;  Oct.  2,  1916,  $700.  What  will  be  due  on  Dec.  31, 
1916? 

7.  Find  the  amount  due  Feb.  24,  1917  on  a  note  for 
$3000,  dated  March  12,  1915,  with  interest  at  7%,  upon 
which  the  following  payments  were  made :  Aug.  18, 
1915,  $235;  April  9,  1916,  $80;  July  3,  1916,  $400; 
Dec.  5,  1916,  $175. 

8.  What  is  the  balance  due  on  Jan.  1,  1917  on  a  note 
for  $500,  dated  July  15,  1916,  bearing  interest  at  5% 
and  having  the  following  indorsements  :  Aug.  20,  1916, 
$27.50;  Oct.  8,  1916,  $125;  Nov.  12,  1916,  $110; 
Dec.  11,  1916,  $65? 

References  374-375. 
Use  the  Merchants'  Rule. 

62.  1.  A  note  of  $850  was  dated  May  25,  1914, 
interest  at  6%.  It  was  indorsed  Aug.  13,  1914,  $50 ; 
Nov.  7,  1914,  $324.95.     What  was  due  March  25,  1915  ? 

2.  A  note  for  $1250,  dated  Jan.  25,  1916,  interest  at 
8%,  bears  the  following  indorsements  :  March  10,  1916, 
$462.50;  Aug.  4,  1916,  $100;  May  22,  1917,  $556, 
What  was  due  on  Jan.  1,  1918  to  settle  the  note  in  full  ? 


RATIONAL  ARITHMETIC  85 

3.  On  a  note  for  $550,  dated  Feb.  5,  1913,  interest 
at  6%,  the  following  payments  were  made :    Oct.   17, 

1913,  $66.10;    March  5,   1914,  $140.     What  was  due 
Nov.  11,  1914.^ 

4.  A  note  for  $2000  was  dated  Dec.  12,  1915,  interest 
at  7%.  It  was  indorsed  as  follows :  June  19,  1916, 
$200;  Dec.  6,  1916,  $338;  Aug.  21,  1917,  $276.50; 
Sept.  12,  1917,  $60.     What  was  due  Oct.  15,  1917.^ 

5.  I  gave  my  note  for  $1080  with  interest  at  5%  on 
Jan.  25,  1914.  I  made  the  following  payments : 
Mar.  1,  1914,  $364.40  ;  May  13,  1914,  $341.50  ;  Sept.  1, 

1914,  $205.     What  was  due  on  settlement,  Jan.   25, 
1915? 

6.  A  note  for  $1500  was  dated  May  11,  1913,  bearing 
interest  at  7.2%.  The  following  payments  were  made : 
Feb.   14,   1914,  $150;    Sept.  23,   1914,  $300;    July  8, 

1915,  $100;   May  29,  1916,  $200.     What  was  due  Sep- 
tember 4,  1916? 

7.  On  a  note  for  $1120,  dated  August  7,  1914, 
interest  at  7%,  payments  were  made  as  follows :  Sept. 
13,  1914,  $80;  Nov.  7,  1914,  $200;  Sept.  15,  1915, 
$450.     What  was  due  Aug.  7,  1916? 

8.  The  following  pa^mients  were  made  on  a  note 
for  $580,  dated  Oct.  17,  1915,  bearing  interest  at  5% : 
Aug.  5,  1916,  $52.50;  April  17,  1917,  $49.30;  Aug.  5, 
1917,  $250.     What  was  due  Sept.  9,  1917? 


BANK  DISCOUNT 


Study  376  to  381  inclusive. 
Reference  38*2. 

63.    Find  the  bank  discount  and  net  proceeds 


Face 

1.  $^40 

2.  $300 

3.  $1000 

4.  $400 

5.  $250 

6.  $350 

7.  $500 

8.  $850 

9.  $600 

10.  $375 

11.  $460 

12.  $2500 

13.  $36500 

14.  $845 

15.  $280 

16.  $430 

17.  $375 

18.  $3000 

19.  $575 

20.  $490 

21.  $450 

22.  $340 

23.  $500 

24.  $1500 

25.  $475 


Date  Time 

Jan.  3,  1916  60  da. 

Sept.  8,  1916  2  mo. 

June  1,  1915  90  da. 

May  1,  1916  3  mo. 

June  1,  1916  3  mo. 

Dec.  30,  1914  4  mo. 

Apr.  3,  1915  6  mo. 

June  1,  1916  6  mo. 

Mar.  3,  1915  90  da. 

Feb.  3,  1916  60  da. 

May  9,  1914  3  mo. 

July  1,  1915  6  mo. 

Mar.  3,  1916  4  mo. 

Jan.  3,  1916  90  da. 

Sept.  8,  1914  30  da. 

Nov.  9,  1915  3  mo. 

Oct.  12,  1916  2  mo. 

Jan.  10,  1916  4  mo. 

Dec.  4,  1915  3  mo. 

Aug.  8,  1915  60  da. 

Mar.  3,  1916  90  da. 

May  2,  1915  6  mo. 

July  5,  1916  60  da. 

Dec.  3,  1914  4  mo. 

Jan.  4,  1916  3  mo. 

86 


Date  of  Disc. 

Jan.  4,  1916 
Oct.  1,  1916 
June  7,  1915 
June  1,  1916 
Aug.  3,  1916 
Jan.  2,  1915 
May  15,  1915 
Oct.  14,  1916 
Apr.  4,  1915 
Feb.  20, 1916 
May  12,  1914 
Aug.  30,  1915 
June  30,  1916 
Feb.  8,  1916 
Sept.  10,  1914 
Nov.  20,  1915 
Nov.  13,  1916 
Jan.  12,  1916 
Jan.  29,  1916 
Sept.  18,  1915 
Apr.  5,  1916 
Aug.  3,  1915 
July  10,  1916 
Feb.  9,  1915 
Jan.  20,  1916 


Rate  op 
Disc. 

6%. 

7%. 


5%. 
6%. 


0- 


^0- 

4%. 

n%. 

41%. 


5%. 
6%. 


'0- 
710/ 

'2/0- 

0- 

7%. 


7%. 


/o- 
7i%. 
4%. 
5%. 


0- 


RATIONAL  ARITHMETIC 


87 


Reference  383. 


64.    Find  the  bank  discount  and   proceeds    of   the 
following  interest-bearing  notes  : 


Face 

1.  $800 

2.  $400 

3.  $2240 

4.  $480 

5.  $1530 

6.  $285 

7.  $390 

8.  $2500 

9.  $460 

10.  $1400 

11.  $2150 

12.  $580 

13.  $490 

14.  $3400 

15.  $1560 

16.  $780 

17.  $?40 

18.  $1375 

19.  $650 

20.  $425 

21.  $730 

22.  $260 

23.  $475 

24.  $390 

25.  $1575 


Date  Time 

Sept.  1, 1915  3  mo. 

June  15, 1916  2  mo. 

Jan.  10,  1916  90  da. 

Jmie  1, 1916  6  mo. 

Mav4, 1916  60  da. 

Sept.  3, 1916  4  mo. 

Feb.  6, 1916  30  da. 

Mar.  10, 1916  3  mo. 

June  12, 1916  4  mo. 

Aug.  3, 1916  90  da. 

July  5, 1916  60  da. 

Mavl2, 1916  6  mo. 

June  6, 1916  30  da. 

Dec.  2, 1916  3  mo. 

Mar.  3, 1916  5  mo. 

Aug.  1, 1915  4  mo. 

July  11, 1916  90  da. 

Mar.  18, 1916  30  da. 

June  19. 1916  5  mo. 

Jan.  31, 1916  1  mo. 

Oct.  6, 1916  2  mo. 

Jan.  14, 1916  5  mo. 

May  10, 1916  90  da. 

Apr.  5, 1916  30  da. 

July  1, 1916  1  mo. 


'0 

5% 
6% 


5% 


Date       Date  of  Disc. 

6%  Sept.  11, 1915 
6%  June  30,  1916 
7%  Feb.  8,  1916 
5%  Aug.  3, 1916 
May  4, 1916 
Sept.  20,  1916 
Feb.  6, 1916 
Mar.  15, 1916 
Aug.  2, 1916 
Sept.  1,  1916 
Julv  7, 1916 
May  12, 1916 
Junes,  1916 
Jan.  3, 1917 
Mar.  15,  1916 
Aug.  11, 1915 
Aug.  1, 1916 
Mar.  24, 1916 
Aug.  1,  1916 
Feb.  4, 1916 
Oct.  6, 1916 
Mar.  3,  1916 
May  11, 1916 
/o     Apr.  12, 1916 
7i%  July  5,  1916 


'0 

6% 

5% 
4% 


0 

5% 
6% 
4% 

o% 
7% 


^0 

10% 


Rate  of 
Disc. 


'0- 

5%. 


0- 


0- 


4%. 
5%. 


'0- 

7%. 


^0- 

6%. 

5%. 
6%. 
7%. 
5%. 


7%. 

7%. 
6%. 
5%. 
6%. 


References  384-385. 

65.  For  what  sum  must  I  write  my  note  in  order  to 
yield  the  following  proceeds  if  discounted  on  the  date 
of  the  note  ? 


88  RATIONAL  ARITHMETIC 


Proceeds 

Time 

Rate 

1. 

$385 

90  da. 

6% 

2. 

$450 

3  mo. 

5% 

3. 

$1285 

60  da. 

8% 

4. 

$370 

4  mo. 

7% 

5. 

$260 

90  da. 

8% 

6. 

$580 

2  mo. 

4i% 

7. 

$290 

6  mo. 

8% 

8. 

$365 

4  mo. 

7% 

9. 

$290 

90  da. 

6% 

10. 

$460 

30  da. 

8% 

11. 

$1340 

2  mo. 

7% 

12. 

$360 

3  mo. 

6% 

13. 

$1500 

1  mo. 

4% 

14. 

$2560 

5  mo. 

7i% 

15. 

$775 

60  da. 

6% 

16. 

$550 

30  da. 

7% 

17. 

$1250 

2  mo. 

8% 

18. 

$875 

4  mo. 

8% 

19. 

$1360 

90  da. 

6% 

20. 

$728  ■ 

60  da. 

5% 

21. 

$450 

5  mo. 

7% 

22. 

$1385 

30  da. 

8% 

23. 

$3760 

60  da. 

6% 

24. 

$1485 

3  mo. 

5% 

25. 

$960 

90  da. 

8% 

RATIONAL  ARITHMETIC  89 


COMPOUND   INTEREST 

Study  386  to  388  inclusive. 

Reference  388. 

66.    Find  the  compound  interest '. 

Principal 

Rate 

Time 

Compounded 

1. 

$7800 

6% 

2  yr. 

Annually 

2. 

$4600 

5% 

2yr. 

Semi-annually 

3. 

$8400 

8% 

Syr. 

Annually 

4. 

$9000 

4% 

Syr. 

Quarterly 

5. 

$3500 

6% 

4  yr.  5  mo. 

Semi-annually 

6. 

$4650 

6% 

1  yr.  8  mo. 

Quarterly 

7. 

$3865 

4% 

1  yr.  2  mo.  15  da. 

Quarterly 

8.  A  note  for  $495.60,  dated  June  10,  1912,  and 
drawing  interest  at  6%  per  annum,  compounded  semi- 
annually, was  paid  March  22,  1916.  What  was  the 
amount  due,  if  no  payments  of  either  interest  or  prin- 
cipal had  been  made  ? 

9.  What  amount  will,  on  June  30,  1917,  discharge 
a  note  of  $3560,  dated  Dec.  1,  1914,  and  drawing  in- 
terest at  8%  per  annum,  compounded  quarterly,  no 
previous  payments  having  been  made  ? 

10.  What  is  the  amount  due  April  1,  1916,  upon  a 
note  for  $480.50,  dated  May  10,  1911,  and  drawing 
interest  at  8%  per  annum,  compounded  semi-annually, 
no  previous  payments  having  been  made  ? 

11.  A  young  man  deposited  $200  in  a  savings  bank 
which  paid  4%  per  annum,  compounded  quarterly. 
If  nothing  was  withdrawn,  what  amount  was  to  his 
credit  at  the  end  of  the  third  year  ? 


90 


RATIONAL  ARITHMETIC 


12.  For  the  benefit  of  his  son  who  is  12  years  old, 
Mr.  A  deposited  in  a  savings  bank  $1000  at  4%,  in- 
terest compounded  semi-annually.  How  much  should 
the  son  receive  when  he  becomes  21  years  old? 


PERIODIC   INTEREST 
Study  389  to  390  inclusive. 

67.    Find  the  periodic  interest : 


1. 
2. 
3. 
4. 
5. 
6. 
7. 


Principal 

$5500 

$450 

$3000 

$2850 
$4650 
$956 

$380 


Rate  per 
Annum 


6% 


0 


'0 

7% 
4% 


0 


0 


Time 

4  yr. 
3yr. 

4  yr. 

5  yr.  3  mo. 

1  yr. 

2  yr.  10  mo. 


Interest  Due 

Annually 

Annually 

Semi-annually 

Annually 

Quarterly 

Quarterly 

Semi-annually 


4  yr.  1  mo. 

8.  What  amount  will  be  due  Feb.  1,  1922,  on  a  note 
of  $3000,  dated  Jan.  1,  1920,  and  drawing  interest  at 
6%  per  annum,  payable  semi-annually,  if.  the  first  four 
interest  payments  are  paid  when  due,  and  no  subse- 
quent payments  made  ? 

9.  What  amount  was  due  July  15,  1917,  on  a  note 
of  $4600,  dated  March  13,  1913,  drawing  interest  at  5% 
per  annum,  payable  semi-annually,  no  previous  pay- 
ments having  been  made  ? 

10.  No  interest  having  been  previously  paid,  what 
was  the  amount  of  a  note  of  $1400  at  6%,  interest  pay- 
able quarterly,  dated  Jan.  1,  1914,  and  paid  Feb.  1, 
1916? 


RATIONAL  ARITHMETIC  91 

11.  What  sum  was  due  Jan.  28,  1917,  on  a  note  of 
$4000,  dated  May  18,  1913,  and  drawing  interest  at  5% 
per  annum,  payable  semi-annually ;  no  payments  hav- 
ing been  made  previous  to  that  time  ? 

12.  A  merchant  bought  a  store  building  for  $9000, 
giving  his  note  without  interest,  payable  2  years  from 
date,  and  8  separate  non-interest-bearing  notes  for  the 
quarterly  interest  at  6%  per  annum.  If  nothing  was 
paid  until  the  maturity  of  the  note,  what  was  the 
amount  then  due  ? 

13.  I  purchased  a  $1250  mortgage  on  which  interest 
at  6%  was  due  semi-annually  on  Jan.  15  and  July  15. 
Owing  to  the  fact  that  no  interest  had  been  paid  since 
Jan.  15,  1917,  I  secured  the  mortgage  at  less  than  its 
face  value.  On  Oct.  27,  1919,  I  made  arrangements 
with  the  mortgagor  whereby  he  paid  the  interest  in 
full  and  $500  on  the  face  of  the  mortgage.  How  much 
did  I  receive  in  all  ? 

14.  What  interest  is  due  Jan.  7,  1920,  on  $875  from 
Nov.  13,  1917,  at  6%,  interest  due  quarterly  and  none 
having  been  paid  ? 

15.  What  is  the  total  interest  on  $386.40  from  Dec. 
31,  1914,  to  Sept.  1,  1919,  interest  due  annually  and 
none  having  been  paid  ?     Rate  4^%. 

16.  What  amount  was  due  Jan.  8,  1920,  on  a  note  of 
$2340  dated  Sept.  1,  1917,  drawing  interest  at  6%,  in- 
terest payable  semi-annually,  if  the  first  two  payments 
were  made  when  due  and  no  subsequent  payments 
made  ? 


AVERAGE  ACCOUNTS 


Study  391  to  396  inclusive. 
Reference  397. 

68.   Average  the  following : 

1.   Dr.  Harold  Chute 


1916 
Oct.    12 
Dec.  20 
1917 
Jan.      5 
Mar.    2 


$  67.85 
71.15 

143.50 
116.20 


Cr. 


1917 

Jan.    10 

$316.20 

Feb.   19 

415.23 

Mar.  24 

99. 

May  10 

271. 

2.    Dr. 

Fred  Ellis 

Cr. 

■ 

1917 

Apr.  18 

$367.40 

May    6 

572. 

May  23 

923. 

June    2 

134.50 

3.   Dr, 

Benjamin  Jones 

Cr. 

92 


4.    Dr. 


RATIONAL  ARITHMETIC 

Howard  Colson 


93 
Cr, 


1916 

Dec.  1 

$540. 

Dec.  15 

236.10 

1917 

Jan.  2 

200. 

Jan.  31 

150. 

5.    Dr, 


James  Mullaney 


Cr. 


1917 

Feb.  5 

$1050.10 

Mar.  10 

826. 

May  1 

924. 

May  31 

186. 

Reference  398 


69.    Average  the  following  : 

1.    Find  cash  balance  on  April  18,  1915. 


Dr. 


Paul  Duncanson 


Cr, 


1915 

Jan.  27 

30  da. 

$420. 

Feb.  17 

10  da. 

300. 

Mar.  1 

20  da. 

540. 

Apr.  12 

30  da. 

600. 

94 


RATIONAL  ARITHMETIC 


2.    Find  cash  balance  on  Jan.  1,  1917. 


Dr. 


Arthur  Bennett 


Cr. 


1916 

Oct. 

17 

15  da. 

$432. 

Nov. 

20 

2  mo. 

864. 

Nov. 

30 

30  da. 

286. 

Dec. 

19 

10  da. 

627. 

3.    Find  cash  balance  on  Dec.  31,  1916. 


Dr 


C.  D.  Adams 


4.    Find  cash  balance  on  Sept.  7,  1916. 


Cr, 


1916 

Aug.  9 

30  da. 

$234. 

Sept.  15 

60  da. 

562. 

Nov.  29 

10  da. 

52.96 

Dec.  21 

15  da. 

715. 

Dr. 


George  Duncan 


Cr, 


1916 

May  6 

30  da. 

$128. 

June  30 

20  da. 

126. 

July  19 

10  da. 

213.20 

Sept.  3 

30  da. 

185. 

RATIONAL  ARITHMETIC 


95 


5.    Find  cash  balance  on  May  1,  1915. 
Dr.  Paul  Jones 


70. 
1.    Dr. 


References  399-400. 


A.  C.  Davis 


When  is  the  above  due  by  average  ? 
What  was  the  cash  balance  Apr.  15,  1917.^ 


CV. 


1915 

• 

Jan.      5 

10  da. 

$400. 

Jan.    31 

30  da. 

90.60 

Mar.     8 

10  da. 

150. 

Apr.   25 

2  mo. 

86.12 

Cr. 


1917 

1917 

Jan. 

1 

2  mo. 

$600. 

Feb.     1 

Cash 

$200. 

Feb. 

2 

30  da. 

240. 

Mar.  18 

Cash 

150. 

Apr. 

6 

10  da. 

360. 

Apr.     3 

Cash 

75. 

2. 

Dr. 

J.  F.  Howard 

Cr, 

1916 

1916 

May 

18 

60  da. 

$209.70 

June     1 

Cash 

$100. 

June 

3 

30  da. 

180. 

June  30 

Cash 

50. 

July 

10 

15  da. 

750. 

July   19 

Cash 

300. 

Aug. 

1 

10  da. 

280.50 

When  is  the  above  due  by  average  ? 
What  was  the  cash  balance  Aug.  21,  1916? 


96 


RATIONAL  ARITHMETIC 


3.    Dr, 


George  Stevens 


When  is  the  above  due  by  average  ? 
What  was  the  cash  balance  July  1,  1915.'^ 


4.    Dr. 


Fred  Ellis 


When  is  the  above  due  by  average  .^ 
What  was  the  cash  balance  May  5,  1917.'^ 


Cr. 


1915 

1915 

Jan. 

20 

2  mo. 

$219.50 

Feb. 

25 

Cash 

$  50. 

Feb. 

25 

30  da. 

218.75 

Mar. 

31 

Cash 

75. 

Mar. 

28 

10  da. 

413. 

Apr. 

30 

Cash 

200. 

June 

30 

10  da. 

216. 

Cr. 


1917 

1917 

Jan. 

1 

10  da. 

$600. 

Feb.  28 

Cash 

$400. 

Feb. 

2 

10  da. 

200. 

Mar.  31 

Cash 

100. 

Mar. 

3 

10  da. 

350. 

Apr.  30 

Cash 

150. 

5.    Dr. 


William  Walker 


Cr. 


1915 

1915 

Jan. 

31 

2  mo. 

$540. 

Feb.  15 

Cash 

$225. 

Feb. 

15 

60  da. 

450. 

Mar.  1 

Cash 

345. 

Mar. 

30 

10  da. 

306.50 

Mar.  10 

Cash 

295. 

When  is  the  above  due  by  average  ? 
What  is  the  cash  balance  Julv  3,  1915  ? 


RATIONAL  ARITHMETIC 


97 


6.    D 


r. 


Benjamin  Brown 


Cr. 


1916 

1916 

Apr.      2 

10  da. 

$150. 

June  25 

Cash 

$300. 

Mav     1 

15  da. 

540. 

Julv  31 

Cash 

360. 

June     3 

10  da. 

450. 

Aug.  10 

Cash 

250. 

July      2 

10  da. 

323. 

When  is  the  above  due  by  average  ? 
What  is  the  cash  balance  Aug.  29,  1916? 


7.    Dr. 


Charles  Smith 


Cr. 


1916 

1916 

Sept.     1 

2  mo. 

$315.60 

Oct.      5 

Cash 

$200. 

Oct.    25 

60  da. 

419.10 

Nov.    1 

Cash 

150. 

Nov.  16 

10  da. 

216.05 

Dec.     2 

Cash 

375. 

When  is  the  above  due  by  average  '^ 
What  was  the  cash  balance  Mar.  5,  1917  ? 


8.    D 


r. 


Robert  Brown 


Cr. 


1915 

1915 

Jan.    2 

1  mo. 

$1800. 

Feb.   18 

Cash 

$300. 

30 

10  da. 

600. 

Feb.  27 

Cash 

300. 

Mar.    5 

Cash 

300. 

When  is  the  above  due  by  average  ? 
What  was  the  cash  balance  Mar.  10,  1915  ? 


98 


RATIONAL  ARITHMETIC 


9.    Dr. 


E.    BOWDOIN 


Cr. 


1916 

1916 

Oct. 

1 

30  da. 

$350. 

Oct.  21 

Cash 

$300. 

Nov. 

8 

10  da. 

340. 

Nov.  24 

Cash 

300. 

Dec. 

9 

15  da. 

210. 

1917 

Jan. 

20 

10  da. 

116. 

When  is  the  above  due  by  average  ? 
What  was  the  cash  balance  Feb.  1,  1917? 


10. 

Dr. 

Sidney 

Berry 

Cr. 

1915 

1915 

June 

1 

60  da. 

$410. 

Aug.  1 

Cash 

$300. 

July 

5 

30  da. 

135. 

Aug.  31 

Cash 

200. 

Aug. 

1 

10  da. 

216.39 

Sept.  4 

Cash 

100. 

Aug. 

31 

15  da. 

162.54 

When  is  the  above  due  by  average  ? 
What  was  the  cash  balance  Oct.  6,  1915  ? 


TAXES 

The  general  principles  of  percentage  are  used  in  figuring  taxes. 
Study  401-404  inclusive. 

References  242-*263  inclusive. 

71.  1.  What  is  the  tax  on  property  assessed  for 
$17,400,  the  rate  of  taxation  being  |%  ? 

2.  What  is  the  tax  on  property  assessed  for  $8500, 
rate  of  taxation  being  16f  mills  on  the  dollar? 

3.  What  is  the  tax  on  property  assessed  for  $23,500, 
the  rate  of  taxation  being  $19.20  on  the  thousand  .^^ 

4.  What  is  the  tax  on  property  assessed  for  $7588, 
the  rate  of  taxation  being  $1.20  on  the  hundred.^ 

5.  I  own  real  estate  worth  $19,500  upon  which  I  pay 
a  tax  at  the  rate  of  $21.40  a  thousand.  I  also  pay  an 
income  tax  of  6%  on  a  net  taxable  income  of  $1400  and^ 
a  poll  tax  of  $2.     What  is  nay  entire  tax.^ 

6.  My  real  estate  is  assessed  at  $6500,  my  personal 
property  at  $1570 ;  my  net  taxable  income  is  $2400. 
Tax  on  the  tangible  property  is  levied  by  the  city  at 
the  rate  of  $19.40  a  thousand ;  an  income  tax  is  levied 
by  the  state  at  the  rate  of  3%  ;  my  poll  tax  is  $2.  What 
is  my  entire  tax  ? 

7.  The  assessed  value  of  real  estate  in  a  town  is 
$1,869,000;    personal    property   is    $2,450,000.     It    is 

99 


100  RATIONAL  ARITHMETIC 

necessary  to  raise  by  taxation  $412,560.     What  would 
be  the  rate  a  thousand  if  there  are  1742  polls  at  $2  each  ? 

8.  In  a  town  whose  valuation  is  $25,000,000,  there 
is  an  increase  in  the  budget  to  cover  additional  expenses 
of  the  public  schools  amounting  to  $40,000.  How  many 
cents  a  thousand  is  the  tax  increased  thereby?  How 
much  will  the  improvement  cost  a  citizen  who  is  worth 
$30,000  ? 

CUSTOMS   AND   DUTIES 

Ad  valorem  duties  are  estimated  according  to  the  value  of  the 

goods  in  conformity  to  the  principles  of  percentage.     Study  405-422 

inclusive. 

References  242-263  inclusive. 

Values  of  units  of  foreign  currency  expressed  in  United  States 
money  will  be  found  in  468. 

72.  1.  What  is  the  ad  valorem  duty  upon  an  im- 
portation valued  at  £430,  Ss,  9d,  allowing  10%  for 
breakage,  duty  being  at  25%  ? 

2.  Find  the  ad  valorem  duty  on  an  invoice  of 
15,834  marks  at  23%,. 

3.  Find  the  ad  valorem  duty  on  an  invoice  of 
3446.18  francs,  duty  being  4^3%. 

4.  Find  the  ad  valorem  duty  on  a  bill  of  1475 
pesos  if  the  duty  is  24%. 

6.  What  is  the  specific  duty  on  13  tons  of  tan  bark 
on  which  there  is  a  duty  of  3  cents  a  hundred  pounds  ? 

6.  What  is  the  specific  duty  on  an  invoice  amount- 
ing to  $760,   allowing    10%  for  breakage,  duty  being 

at  35%  ? 


RATIONAL  ARITHMETIC  101 

7.  Find  the  duty  at  10  cents  a  square  yard  and 
40%  ad  valorem  on  a  rug  12'X18',  imported  from  Eng- 
land and  invoiced  at  £14. 

8.  What  is  the  total  duty  on  140  cases  of  plate 
glass,  each  containing  25  plates,  20''x48''  at  8^  a 
square  foot  ? 

9.  What  is  the  duty  on  an  invoice  of  2300  yards  of 
27-inch  goods,  invoiced  at  8^  9^  a  yard,  subject  to  an 
ad  valorem  duty  of  40%  and  a  specific  duty  of  6^  a 
square  yard  ? 

10.  What  is  the  duty  at  60%  on  a  bill  amounting 
to  £736  9s  Sd  ? 

11.  W^hat  is  the  duty  at  30%  ad  valorem  on  U\o 
bales  of  burlap,  each  bale  containing  40  webs,  each 
web  being  48  yd.  long  and  30  in.  wide,  invoiced  at  30^ 
per  square  yard  ? 

12.  What  is  the  duty  at  20^  per  square  yard  and 
35%  ad  valorem  on  1750  yards  of  cloth  invoiced  at 
7  francs  per  yard  ? 

13.  A  merchant  imported  a  lot  of  steel  knives  from 
England  as  follows:  75  doz.  at  12^  Qd;  50  doz.  at  18.? 
6d\  30  doz.  at  £l  5s  Qd;  20  doz.  at  £l  8^  6d;  12  doz. 
at  £2  9s  6d;  10  doz.  at  £2  10^  6d.  The  charges  in 
England  amovmt  to  £7  12^*  Qd.  The  consul's  fee  was 
125  6d.  Marine  insurance  was  20  (^  per  hundred  on  the 
value  of  the  invoice.  The  cartage  amounted  to  $2.50. 
The  duty  was  30%  ad  valorem  and  30  p  per  dozen. 
Find  the  total  cost  of  the  invoice. 


INSURANCE 

Study  423-434  inclusive. 
References  M^-263  inclusive. 

73.  1.  My  house  is  insured  for  $4500  for  a  period 
of  five  years  at  21%.     What  is  the  premium  ? 

2.  A  merchant  insured  his  stock  of  goods  for  $5600 
at  the  rate  of  li%  per  annum.  What  annual  premium 
does  he  pay  ? 

3.  A  factory  is  insured  for  $1 '25, 000  in  four  com- 
panies. A  carries  i  of  the  insurance,  B  carries  i, 
C  carries  i,  and  D  i.  A  fire  occurs  causing  a  damage 
of  $50,000.  For  how  much  will  each  company  be 
responsible  ?  • 

4.  A  stock  of  goods  is  insured  in  four  companies  as 
follows  :  $1500  in  A,  $2400  in  B,  $3200  in  C,  and  $2500 
in  D.  The  goods  are  damaged  to  the  extent  of  $8000. 
How  much  should  each  company  pay.^ 

5.  A  building  worth  $85,000  was  insured  for  $68,000, 
and  afterwards  damaged  by  fire  to  the  extent  of  $4500. 
The  policy  contains  the  average  clause.  What  amount 
of  insurance  can  be  collected  from  the  company  ? 

6.  A  vessel  worth  $50,000  is  insured  for  $20,000  in 
company  A  and  $18,000  in  company  B.  The  vessel 
is  damaged  to  the  extent  of  $20,000.  What  amount 
is  to  be  paid  by  each  company  ? 

102 


RATIONAL  ARITHMETIC  103 

7.  I  insured  my  building  worth  $80,000  for  80%  of 
its  value  at  f%  premium  with  the  iEtna  Insurance 
Company.  The  iEtna  Insurance  Company  later  re- 
insured $20,000  in  the  Niagara  Insurance  Company 
and  $18,000  in  the  Massachusetts  Fire  and  Marine 
Insurance  Company.  The  property  is  damaged  to 
the  extent  of  $30,000.  What  was  the  net  loss  to  each 
of  the  companies  ? 

8.  I  have  a  policy,  containing  the  average  clause, 
for  $7500  on  merchandise  in  stock  worth  $9000, 
upon  which  I  have  paid  a  premium  of  |%.  A  fire 
occurs  by  which  the  goods  are  damaged  to  the  ex- 
tent of  $4000.  What  was  my  total  loss  and  the  net 
loss  to  the  company  ? 

LIFE   INSURANCE 

Study  435-441  inclusive. 
Reference  442, 

74.  1.  What  would  be  the  annual  premium  on  a 
policy  for  $2500,  premiums  payable  annually  during 
life,  at  the  age  of  21  years  ?  26  years  ?  32  years  ? 
38  vears  ? 

2.  What  would  be  the  annual  premium  on  a  fifteen - 
year  endowment  policy  for  $5000,  at  the  age  of  23 
years  ?     27  years  ?     32  years  ?     37  years  ? 

3.  What  would  be  the  annual  premium  on  a  policy 
for  $4000,  premiums  due  annually  for  a  period  of  ten 
years,  policy  payable  at  death  only,  at  the  age  of 
20  vears  ?     25  vears  ?     35  vears  ? 


104  RATIONAL  ARITHMETIC 

4.  What  would  be  the  annual  premium  on  a  twenty- 
year  endowment  policy  for  $6500,  age  of  insured  at 
nearest  birthday  23  years  ?     29  years  ?     37  years  ? 

5.  What  would  be  the  annual  premium  on  an  or- 
dinary life  policy  for  $3500,  premiums  to  be  paid 
annually  for  twenty  years,  policy  to  mature  at  death, 
age  of  insured  at  nearest  birthday  28  years  ?  23  years  ? 
38  years  ? 

6.  A  man  insured  his  life  at  the  age  of  23  years  on  a 
twenty-year  endowment  plan,  payments  to  be  paid 
annually,  amount  of  policy  $5000.  He  died  at  the  age 
of  33  years.  How  much  less  would  he  have  paid  in 
premiums  if  he  had  been  insured  by  the  ordinary  life 
plan? 

7.  A  man  at  the  age  of  35  took  out  a  fifteen -year 
endowment  policy  for  $2000.  What  annual  premium 
must  he  pay  ?  He  lives  20  years  and  receives  the  face 
of  the  policy.  How  much  less  will  this  amount  to 
than  it  would  have  if  he  had  invested  the  premium 
at  4%  compound  interest  "^ 

8.  A  man  at  the  age  of  25  took  out  a  $3000  twenty 
payment  life  policy.  He  died  after  paying  ten  pre- 
miums. What  was  the  annual  premium  ?  How  much 
more  did  his  family  receive  than  the  premiums 
amounted  to,  making  no  allowance  for  interest  ? 

9.  Three  men,  aged  24,  take  a  policy  for  $1000  each. 
One  takes  an  ordinary  life  policy,  one  a  twenty-year 
life  policy,  and  one  a  twenty -year  endowment  policy. 
At  the  end  of  five  years  how  much  had  each  paid  in 
premiums  ? 


EXCHANGE 

DOMESTIC   EXCHANGE 
TO   FIND   THE   VALUE   OF   A   SIGHT   DRAFT 

Study  443-449  inclusive. 
References  255-258  inclusive. 

75.  Find  the  value  of  the  following  drafts : 

1.  $2300  bought  at  i%  discount. 

2.  $1400  bought  at  lf%  premium. 

3.  $1740  sold  at  1%  premium. 

4.  $3000  bought  at  li%  discount. 

5.  $2450  sold  at  li%  premium. 

6.  $4500  bought  at  $1.50  premium. 

7.  $1240  sold  at  $1.25  discount. 

8.  $1450  bought  at  $.50  premium. 

9.  $4300  sold  at  |%  discount. 
10.  $9000  sold  at  i%  discount. 

TO   FIND   THE   VALUE   OF   A   TIME   DRAFT 

76.  To  find  the  cost  or  selling  price  of  a  time  draft : 
Find  the  net  proceeds  of  the  draft  according  to  the 
principles  of  bank  discount  (381-382).  From  this 
deduct  the  exchange  discount,  or  to  it  add  the  exchange 
premium,  found  as  in  75. 

105 


106  RATIONAL  ARITHMETIC 

77.  What  is  the  cost  of  a 

1.  60-day  draft  for  $6000,  |%  premium,  interest 

at  6%  ? 

2.  30-day  draft  for  $2200,  i%  premium,  interest 

at  7%  ? 

3.  15-day   draft  for  $2500,   f%   discount,   interest 

at  7%  ? 

4.  30-day   draft   for   $750,   |%   premium,   interest 

at  5%  ? 

5.  30-day  draft  for  $5000  at  |%  discount,  interest 

at  6%  ? 

6.  90-day  draft  for  $2300  at  ^%  premium,  interest 
at  4i%  ? 

7.  60-day  draft  for  $2500  at  |%  discount,  interest 
at  4%  ? 

8.  30-day  draft  for  $2350  at  f%  premium,  interest 
at  4i%  ? 

9.  60-day    draft    for    $1240    at    $1.25    premium, 
interest  at  5%  .^ 

10.    30-day  draft  for  $2350  at  $1.50  discount,  interest 
at  6%  ? 

TO   FIND   THE   FACE   OF   A   DRAFT 

78.  Find  the  value"  of  a  draft  of  $1  as  explained  in 
75  and  76,  and  divide  the  given  value  by  this. 

79.  What  is  the  face  of  a  sight  draft  which  can  be 
bought  for 

1.  $1207.50  if  exchange  is  at  f%  premium  ? 

2.  $1091.75  if  exchange  is  at  f%  discount  .^^ 


RATIONAL  ARITHMETIC  107 

3.  $2453.28  if  exchange  is  at  i%  premium? 

4.  $1636.02  if  exchange  is  at  f%  discount? 

5.  $4234.62  if  exchange  is  at  i%  discount? 

80.  What  is  the  face  vakie  of  a  30-day  draft  which 
can  be  bought  for 

1.  $1183.50  at  i%  discount,  interest  6%  ? 

2.  $1453.84  at  i%  premium,  interest  5%  ? 

3.  $2493.62  at  $1.20  premium,  interest  4i%  ? 

4.  $3977.33  at  $1.50  discount,  interest  5%? 

5.  $2843.55  at  f%  premium,  interest  6%  ? 

FOREIGN    EXCHANGE 

Study  450-452  inclusive. 
Reference  468. 

81.  Find  the  exchange  value  of  a  bill  for ' 

1.  £540  at  4.83i. 

2.  £1476  at  4.85f. 

3.  £250  9s  8d  at  4.85^. 

4.  £783  13s  lid  at  4.841. 

5.  15,642  francs  at  5.18|. 

6.  8575.75  francs  at  5.19. 

7.  8462.73  francs  at  5.20^. 

8.  2648.55  francs  at  5.19f. 

9.  1284  marks  at  94^. 

10.  2556  marks  at  95i. 

11.  6742  marks  at  94f. 

12.  1287.5  marks  at  94^. 


108  RATIONAL  ARITHMETIC 

13.  789.7  guilders  at  40i. 

14.  2345  guilders  at  40^. 

15.  1286  guilders  at  401. 

16.  1286.5  guilders  at  39J. 


82.  What  is  the  face  of  an  English  bill  of  exchange 
that  cost 

1.  $2213.88  at  4.85i.^        3.    $585.37  at  4.84f,P 

2.  $6060.95  at  4.84^.^       4.    $1209.38  at  4.83f.? 

83.  What  is  the  face  of  a  French  bill  of  exchange 
that  cost 

5.  $819.29  at  5\19f.?  7.    $225.44  at  5.19  ? 

6.  $2316.04  at  5. 18i.?        8.    $88.42  at  5.201.^ 

84.  What  is  the  face  of  a  German  bill  of  exchange 
that  cost 

9.    $348.60  at  95f?  11.    $393.46  at  95i? 

10.    $2945.31  at  94i?         12.    $13795.19  at  94|.^ 

85.  What  is  the  face  of  a  Dutch  bill  of   exchange 
that  cost 

13.  $226.10  at  40f?  15.    $235.52  at  40i? 

14.  $458.97  at  39i  ?  16.    $6415.60  at  40i? 


STOCKS  AND   BONDS 

The  general  principles  of  percentage  are  involved  in  solving  the 
following  problems  (242-263). 
Studv  453-464  inclusive. 

86.  1.  A  railroad  with  a  capital  stock  of  $2,500,000 
declared  a  dividend  at  the  rate  of  5%.  What  was  the 
total  amount  of  the  dividend.^  How  much  did  A, 
the  owner  of  350  shares,  receive  ? 

2.  What  will  be  the  total  dividend  at  5%,  declared 
by  a  $3,000,000  corporation  ? 

3.  A  manufacturing  corporation  with  a  capital  of 
$50,000  levies  an  assessment  of  8%  upon  its  stock- 
holders. What  is  the  total  assessment,  and  what  will 
B  be  called  upon  to  pay,  if  he  holds  230  shares  of  $100 
each  ? 

4.  What  dividend  would  I  receive  on  163  shares 
of  $100  stock  at  the  rate  of  5%  ? 

5.  A  corporation  with  a  capital  stock  of  $475,500, 
divided  $38,040  among  its  stockholders.  What  was 
the  rate  of  this  dividend  ? 

6.  A  corporation  of  which  I  am  a  stockholder 
declares  a  dividend  of  4^%.  My  dividend  check  is 
$652.50.  How  many  shares,  par  value  of  $100,  do  I 
own? 

109 


110  RATIONAL  ARITHMETIC 

7.  A  mining  corporation  of  which  I  am  a  stock- 
holder declares  a  dividend  of  10%.  I  receive  $125  as 
my  dividend.  How  many  shares  of  $10  par  value  do 
I  own  ? 

8.  A  corporation  with  $1,500,000  capital  had  a 
gross  income  of  $975,000.  Its  total  expenses  were 
$785,000.  Its  directors  set  $100,000  aside  as  a  reserve 
fund ;  the  rest  was  divided  among  the  stockholders. 
What  per  cent  dividend  was  declared  ? 

9.  What  is  the  market  value  of  130  shares,  par  value 
$100,  of  Q.,  O.  &  K.  C.  R.  R.  quoted  at  113^.^ 

10.  What  is  the  market  value  of  640  shares,  par 
value  of  $10  each,  of  New  England  Manufacturing 
Company,  quoted  at  85^  ? 

11.  How  much  must  I  pay  for  75  shares  ($100  par 
value)  B.  &  M.  R.  R.  at  49i,  brokerage  i%? 

12.  What  is  the  cost  of  130  shares  ($100  par  value) 
Bell  Telephone  at  418i,  brokerage  i%  ? 

13.  Find  the  total  cost  of  $1000  L.  &  E.  R.  R.  2d  4's 
at  1021;  $4000  C.  &  N.  W.  5's  at  102f ;  40  shares 
($100  par  value)  of  A.  T.  &  S.  F.  R.  R.  at  43f ;  75 
shares  ($100  par  value)  B.  &  M.  R.  R.  at  34f ;  brokerage 

on  all  i%  ? 

14.  What  is  the  net  cost  of  150  shares  ($100  par 
value)  of  B.  &  A.  R.  R.  at  134f ;  75  shares  ($100  par 
value)  M.  C.  R.  R.  at  104f ;  $5000  S.  E.  L.  Co.  6's  at 

95i ;  brokerage  on  all  i%  ? 

15.  What  is  the  proceeds  of  450  shares  ($50  par 
value)  sold  at  102f ,  brokerage  i%  ? 


RATIONAL  ARITHMETIC  111 

16.  How  much  must  I  invest  in  U.  S.  4's  of  1932 
to  secure  a  quarterly  income  of  $450,  bonds  selling  at 
108i,  brokerage  i%  ? 

17.  I  invested  $3376.25  through  my  broker  at  i% 
commission,  in  U.  S.  4%  bonds  at  115f.  What  will  be 
my  annual  income  ? 

18.  How  much  must  I  invest  in  U.  S.  4's  of  1935  to 
secure  a  quarterly  income  of  $600,  bonds  selling  at 
108f ,  brokerage  i%  ? 

19.  Sold  75  shares  ($100  par  value)  railroad  stock 
through  a  broker  and  received  $7388  net  proceeds. 
At  what  quotation  did  the  broker  sell  the  stock? 

20.  At  what  price  may  6%  stock  be  bought  to  re- 
ceive 5%  on  the  investment,  brokerage  i%? 

21.  What  price  can  I  afford  to  pay  for  7%  bonds  in 
order  to  realize  8%  income  on  the  investment,  broker- 
age i%  ? 

22.  What  price  will  I  pay  for  5%  bonds  bought 
through  a  broker  so  as  to  bring  in  a  net  income  of  4% 
on  the  investment  ? 

23.  At  what  quotation  could  8%  preferred  stock  be 
bought  through  a  broker  to  realize  5%  income  on  the 
investment  ? 

24.  What  price  can  I  afford  to  pay  for  7%  bonds 
bought  through  a  broker  so  as  to  receive  a  net  income  of 
6%  on  the  investment  .^^ 

25.  What  per  cent  income  on  the  investment  will 
be  realized  if  4%  stock  is  bought  at  79|,  brokerage  i%  ? 

26.  What  per  cent  is  realized  on  the  investment  if 
6%  stock  is  bought  at  74 1,  brokerage  i%  ? 


112  RATIONAL  ARITHMETIC 

27.  5%  bonds  bought  at  124 J  would  bring  what  per 
cent  on  the  investment,  brokerage  i%  ? 

28.  Stock  bought  at  79|,  brokerage  i%,  yields  4% 
on  the  investment.     What  is  the  rate  of  dividend  .^ 

29.  Which  is  the  better  investment  and  how  much : 
stock  paying  6%  dividend,  bought  at  74|,  or  stock 
paying  9%  dividend,  bought  at  119J,  brokerage  i%? 

30.  I  bought,  through  a  broker,  52  shares  of  stock 
at  84.  I  paid  an  assessment  of  5%  and  then  sold  them 
at  99f .     How  much  did  I  gain,  brokerage  i%  ? 

31.  I  have  $8000  to  invest.  I  am  offered  bank 
stock  at  375  yielding  3^%  each  three  months,  or  stock 
in  a  shoe  manufacturing  company  at  150  paying  4% 
semi-annually.  I  have  made  up  my  mind  to  invest 
in  the  stock  which  will  give  me  the  greater  dividend. 
Which  shall  I  buy  and  what  will  be  the  total  dividend 
each  year  ? 

32.  By  investing  $18,750  in  150  shares  of  stock  I  am 
able  to  realize  4%  on  the  investment.  What  rate 
of  dividend  does  the  stock  pay  ? 

33.  What  can  I  afford  to  pay  for  8%  stock  to  realize 
5%  on  the  investment  ? 

34.  I  have  been  offered  a  block  of  4v  Libertv  Bonds 
at  98.  What  per  cent  would  they  yield  on  the  invest- 
ment ? 


PART  TWO 


RATIONAL  ARITHMETIC 


PART  TWO 

87.  Arithmetic  is  the  measure  of  values  or  quantities 
expressed  in  figures. 

All  arithmetic  consists  of  increasing  or  decreasing 
values  or  quantities. 

88.  Addition  is  the  simple  or  basic  operation  of 
increasing  values  or  quantities. 

The  sign  of  addition  is  +,  read  plus. 

89.  Subtraction  is  the  simple  or  basic  operation  of 
decreasing  values  or  quantities. 

The  sign  of  subtraction  is  — ,  read  minus. 

90.  Multiplication  is  a  short  method  of  addition  by 
which  quantities  or  values  are  increased  at  a  fixed 
ratio  —  by  a  given  number. 

The  sign  of  multiplication  is  X,  read  times. 

91.  Division  is  a  short  method  of  subtraction  by 
which  a  certain  quantity  or  value  is  reduced  at  a  fixed 
ratio  —  by  a  given  number. 

The  sign  of  division  is  -^ ,  read  divided  by. 

Inasmuch  as  these  operations  are  quite  different  in 
their  applications,  they  are  treated  separately,  and  are 
known  as  the  four  fundamental  operations  of  arithmetic. 

1 


2  RATIONAL  ARITHMETIC 

NOTATION 

92.  For  a  thorough  understanding  of  arithmetic  it 
is  necessary  to  be  famihar  with  the  system  of  notation 
used  in  expressing  values  and  quantities  in  figures. 

Ten  characters  (figures)  are  used,  nine  of  which 
have  a  positive  or  integral  value.  These  are  repre- 
sented by  the  figures  12345678  9.  The  tenth 
figure,  0  (read  cipher^  zero,  or  naught),  represents  nothing 
and  has  no  integral  value.  In  other  words,  the  figure  3 
stands  for  three  individual  units ;  5,  for  five  individual 
units ;  7,  for  seven ;  9,  for  nine ;  while  the  cipher  is 
used  to  visualize  nothing. 

These  nine  integral  units,  with  their  accompanying 
cipher,  are  given  a  distinct  value  according  to  their 
position  in  relation  to  a  fixed  line  represented  by  the 
decimal  point.  Thus,  one  in  the  units  column  —  the 
first  column  to  the  left  of  the  line  —  means  one  whole 
unit. 

Move  this  1  to  the  next  column,  one  place  to  the 
left ;  fill  in  the  space  from  which  it  has  been  taken 
with  a  cipher  to  show  that  nothing  is  there;  it  then 
represents  the  value  of  "  ten  "  and  is  so  read. 

Move  it  one  more  column  to  the  left  and  it  repre- 
sents ten  times  ten,  or  one  hundred,  and  so  on. 

Writing  the  1  in  the  second,  or  tens  column,  and  the 
figure  3  in  the  first,  or  units  column,  we  show  10  units 
(in  the  tens  column)  and  3  units  (in  the  units  column), 
that  is,  13,  read  thirteen. 

93.  It  will  readily  be  seen  that  the  removal  of  a  figure 
one  column  to  the  left  multiplies  its  value  by  ten.     It  will 


RATIONAL  ARITHMETIC 


also  be  apparent  that  bringing  it  back  one  place  to  the 
right  divides  its  value  by  ten. 

This  is  the  governing  principle  of  notation  and  may  be  appHed 
on  either  side  of  the  decimal  hne.  The  figure  1  starting  to  the  left 
of  the  line,  in  the  units  column,  and  moving  one  place  to  the  right 
becomes  yq^  of  1 ;  moved  another  place  to  the  right  it  becomes  -^o  oi 

moved  another  place  to  the  right  it  becomes  j^ 


1 

10' 


w 


hich 


IS 


1 


100  ' 


of 


or 


and  so  on  without  limit. 


10  0'  "^    10  00' 

If  a  student  has  difficulty  in  learning  the  value  of  a  figure  to  the 
right  of  a  decimal  point  (which  is  merely  the  line  of  division  between 
whole  numbers  and  their  fractional  parts  represented  by  tenths),  it 
is  suggested  that  he  take  an  ordmary  sheet  of  writing  paper,  turn  it 
sidewise,  write  the  names  of  the  various  places  in  the  columns  thus 
formed,  draw  a  heavy  line  to  represent  the  decimal  line,  write  the 
decimal  notation  to  the  right  of  this  line,  and  then  place  figures  in 
such  columns  as  may  appeal  to  him,  calling  them  by  the  names  of 
the  values  written  in  the  columns. 


m 

a 
.2 

'C 

o 

-a 
•a 

6 

on 

a 
.2 

« 
3 

en 
C 

4 

C 

i 

-a 

3 

8 

tn 

i 

a; 
H 

9 

5 

tn 

a 

CD 

o 

H 

T3 
OJ 
f- 

'^ 

c 

3 

6 

d 

!D 
O 

d 
a 

3 

«3 

T3 

d 

03 

3 
O 

H 

1 

2 

■ 

■c 

d 
3 

K 

1 
1 

4 

d 

1 

1 

0 
1 

8 

c 
< 

'd 

1 
0 
3 
0 
3 

3 

D 

d 
H 

1 

0 
0 

c 

r 
i 

CO, 

.d 
-o 

1 

1 
0 

Ire 

nil 
lur 

tn 

d 

03 
tc 
3 

2 
H 

1 

dt 
ioi 
idr 

73 

-d 
-tj 
-a 
a 

03 

CO 

3 
O 

-d 

d 
« 
H 

hir 

a, 
ed 

m 

d 

o3 

CO 

3 
O 

-c 

o 

d 

3 

ty- 
six 
eij 

73 

d 

fo 
h 

73 

d 

d 
<u 

H 

iirl 
un( 

y-t 

n 

-d 

-u 

d 

3 
t^ 

d 

3 

hH 
M-l 

Dill 

ire 
hr 

1 

Read  one 

Read  ten 

Read  thirteen 

Read  one  hundred 

Read  one  thousand,  one 
hundred  thirteen 

Read  one-tenth 

Read  one  one-hundredth 

Read  one  one-thousandth 

Read  one  trillion,  six  hun- 
ion,  eight  hundred  ninety-five 
d  thirty-two  thousand,  four 
ee. 

4  RATIONAL  ARITHMETIC 

Exercises  of  this  kind  are  very  valuable  for  students  whose  minds 
are  so  constituted  that  they  have  difficulty  in  reading  decimals. 
Difficulty  in  reading  decimals  should  not  be  ascribed  to  arithmetical 
weakness,  but  rather  to  inability  to  use  the  imagination  in  repre- 
senting figure  pictures. 

94.  Figures  to  the  left  of  the  decimal  line  represent 
whole  numbers  and  are  called  Integers. 

95.  Figures  to  the  right  of  the  decimal  line  represent 
parts  (tenths,  hundredths,  etc.),  and  are  called  Deci- 
mals,  ' 


COMMON  PROCESSES 

ADDITION  —  INTEGERS 

96.  Addition  is  the  process  of  combining  several 
numbers  into  one  quantity  that  shall  equal  the  value 
of  all. 

(a)  Only  numbers  representing  like  values  or  like  quantities,  or 
parts  of  like  values  or  like  quantities,  can  be  added,  thus,  3  cows 
can  be  added  to  2  cows  and  the  result  will  be  7  cows.  .5  horses 
added  to  5  cows  would  produce  10  things,  but  they  would  be 
neither  horses  nor  cows  —  just  10  animals. 

(b)  Figures  not  used  to  measure  value  or  quantity  can  be  added 
as  a  mere  matter  of  counting.  Furthermore,  unlike  things  may  be 
added,  if  first  reduced  to  common  terms.  The  term  "animal"  is 
common  to  both  horses  and  cows. 

97.  The  name  Addend  is  applied  to  any  of  the  in- 
dividual quantities  or  values  that  are  added. 

98.  The  name  Sum  is  applied  to  the  total  value  of 
the  addends.     It  is  the  result  of  addition. 

ILLUSTRATED   SOLUTION 

99.  Problem :  $324.23  +  $89.96  +  $742.05  +  $23  + 
$1.95  +  $796.45=  ? 


RATIONAL  ARITHMETIC 


$324 

23 

89 

96 

742 

05 

23 

1 

95 

796 

45 

$1977 

64 

the  next,  or 
The  sum  of 
total  of  the 
both  figures 


Arrange  the  addends  in  a  column,  placing  decimal 

point  under  decimal  point  so  as  to  form  the  decimal 

line.     Add    either    up    or    down.     Beginning    with 

cents    (first    right-hand    column)    reading    up    and 

combining  values  as  we  go,  we  have  5,  10,  15,  21, 

24  cents.     This  is  2  tens-cents  and  4  units-cents. 

Write   the   4    units-cents    under   the   first   column. 

Carry   2   tens-cents   to   the   next  column.     Adding 

as  before  the  result  is  26.     Write  6,  carry  2.     Adding 

unit-dollars  column,  we  have  27.     Write  the  7,  carry  2. 

the  next  column  is  27.     Write  the  7,  carry  2.     The 

next  column  is  19.     As  this  is  the  last  column  write 


100.  To  Check  the  Work:  Add  each  column  sepa- 
rately, beginning  with  the  first  right-hand  column. 
Write  the  results  as  partial  sums,  one  under  the 
other,  each  one  place  to  the  left  of  its  predeces- 
sor, thus : 


First  column  adds 

24 

Second  column  adds 

2 

4 

Third  column  adds 

25 

Fourth  column  adds 

25 

Fifth  column  adds 

17 

1977.64 


or  we  may  begin  on  the  left  and  work  in  the  opposite 
direction,  adding,  either  up  or  down,  thus : 


First  column  adds 

17 

Second  column  adds 

25 

Third  column  adds 

25 

Fourth  column  adds 

2 

4 

Fifth  column  adds 

24 

1977.64 


RATIONAL  ARITHMETIC  7 

101.  In  billing  it  is  sometimes  desirable  to  add 
quantities  written  in  a  horizontal  line,  thus : 

24  +  112+36+43  +  '246  +  95  =  556 

The  secret  of  accurately  adding  in  this  way  is  to 
add  from  left  to  right  —  because  we  read  from  left  to 
right  and  the  eye  naturally  "  picks  up  "  the  proper 
quantities  in  traveling  this  way,  and  does  not  become 
confused,  as  is  often  the  case  if  we  try  to  add  from 
right  to  left. 

Thus,  in  the  above,  beginning  with  units  in  the  first  left-hand 
quantity,  we  add  with  the  following  results,  4,  6,  12,  15,  21,  2G  : 
write  6  as  units,  carry  2.  In  the  tens  from  left  to  right  we  add  2, 
4,  5,  8,  12,  16,  25  :  write  the  5  as  tens,  carry  2.  In  the  hundreds 
from  left  to  right  we  add  2,  3,  5  :  write  the  5  as  hundreds,  giving 
55Q  as  the  total. 

Note.     For  practice  problems  in  addition  see  pars.  1  to  8  inclusive. 
SUBTRACTION  —  INTEGERS 

102.  Subtraction  is  the  process  of  decreasing  one 
value,  or  quantity,  by  taking  from  it  a  smaller  value, 
or  quantity. 

103.  The  Minuend  is  the  larger  number ;  the  one 
from  which  another  number  is  taken. 

104.  The  Subtrahend  is  the  number  that  is  de- 
ducted from  the  minuend ;  it  is  the  smaller  number 
which  is  taken  out  of  the  larger. 

105.  The  Difference,  or  Remainder,  is  the  number 
that  is  left  after  the  subtrahend  has  been  taken  from 
the  minuend. 


8  RATIONAL  ARITHMETIC 

ILLUSTRATED    SOLUTIONS 

106.  Prohlem:  $2294.18-$346.23=  ? 

1    8  3  Write  the  subtrahend  under  the  minuend  so  that 

$2294  18  the  decimal  points  will  form  the  decimal  line.  Sub- 
346  23  tract:  8  —  3  =  5.  Write  5  in  the  proper  column. 
1^1947  95  Since  2  cannot  be  taken  from  1  we  must  bring  over  1 
from  the  next  column  to  the  left.  We  know  that  one 
in  the  third  column  is  10  of  the  second  column  (par.  93) ;  hence  we 
now  have  11  in  this  column.  11—2  =  9.  Write  9.  In  the  third 
column  we  now  have  3  —  6  which  cannot  be  performed.  Then  take 
1  from  9  in  the  next  column,  which  would  give  us  13  —  6  =  7.  Write 
the  7.  In  the  fourth  column  we  now  have  8—4=4.  Write 
the  4.  In  the  fifth  column  we  now  have  2  —  3  which  cannot  be  per- 
formed. Take  one  from  the  next  column  which  gives  us  12  —  3  =  9. 
Write  the  9.     In  the  last  column  we  have  1—0  =  1.     Write  the  1. 

107.  To  Check  the  Work:  Add  the  subtrahend  and 
the  remainder. 

The  result  should  be  the  minuend,  thus : 

$1947.95  +  $346.23  =  $2294.18 

108.  Another  Method  of  Subtraction.  From  the 
work  used  in  checking,  another  method  for  finding 
the  difference  between  two  quantities  is  derived.  By 
it  we  simply  write  the  figiires  that  must  be  added  to 
the  subtrahend  to  make  it  equal  the  minuend,  thus : 

109.  Pro6/(?m;  $2294.18— $346.23=  ?    iVns.  $1947.95 

Start  with  the  subtrahend,  |346.23.  To  the  first  right-hand 
figure,  3,  add  5  to  make  8,  the  right-hand  figure  of  the  minuend. 
Write  5  in  the  difference.  To  the  second  figure,  2,  add  9,  which 
makes  11;  write  9  in  the  difference.. :  Change  the  third  figure,  6, 
to  7  by  addmg  the  1  carried  from  11.     To  the  newjthirfl  figure,  7^ 


RATIONAL  ARITHMETIC 


9 


add  7  to  make  14.  Write  7  in  the  difference.  Change  the  fourth 
figure,  4,  to  5,  by  adding  the  1  carried  from  14.  To  the  new  fourth 
figure,  o,  add  4  to  make  9.  Write  4  in  the  difference.  To  the  last 
figure,  3,  add  19  to  make  22.     Write  19,  making  1947.95  in  all. 

This    process    is    especially    valuable    in    balancing 
accounts,  thus  : 

Dr.  cash  Cr. 


519 

25 

125 

40 

136 

123 

32 

798.77 

143 

52 

295.22 

46 

50 

Balance 

503 

55 

798 

77 

798 

77 

In  the  above,  add  each  side  of  the  account  separately,  setting 
results  in  small  figures,  on  proper  side.  Then  add  to  the  smaller 
sum  the  figures  required  to  make  it  equal  the  larger,  inserting  these 
figures,  as  Balance.     Then  add  both  sides  as  a  check. 

Note.     For  practice  problems  in  subtraction  see  par.  9. 

MULTIPLICATION  —  INTEGERS 

110.  Multiplication  is  the  process  of  increasing  a 
given  value  or  quantity  a  given  number  of  times. 

111.  Multiplicand  is  the  name  applied  to  the  value 
or  quantity  that  is  increased. 

112.  Multiplier  is  the  name  applied  to  the  number 
representing  the  number  of  times  the  multiplicand  is 
increased. 

113.  Product  is  the  name  applied  to  the  final  result 
of  increasing  the  multiplicand  the  number  of  times 
indicated  by  the  multiplier. 


10  RATIONAL  ARITHMETIC 

114.  Factor  is  a  name  applied  to  either  the  multi- 
plicand or  multiplier,  because  both  are  factors  (or 
makers)  of  the  product. 

ILLUSTRATED   SOLUTIONS 

115.  Problem  :   What  is  the  value  of  6  times  $4682  .'^ 

Begin  with  the  units'  column,  6X2  =  12.     12  is  2 

tJ54Do/i        units  and  1  ten.     Write  the  2  units  in  the  units'  cohimn, 

6        carry   1   ten.     6X8  =  48+1    (carried)  =49  tens,  which 


$28092        is  9  tens  and  4  hundreds.     Write  the  9  in  the  tens' 
column,  carry  4.     6X6  =  36+4  =  40.     Write  0,  carry  4. 
6X4  =  24+4  =  28.     As  this  is  the  last  column,  write  28. 

116.  To  Check  the  Work:  Multiply  each  figure 
separately  beginning  with  the  left  column.  Write  the 
products  under  one  another  for  addition,  but  remov- 
ing each  one  place  to  the  right,  thus : 


6X4  = 

24 

6X6  = 

36 

6X8  = 

48 

6X2  = 

12 

28092 

117.    Problem:   239X$8461=.? 

$8461  First  multiply  8461  by  9  units  as  explained  above. 

239      The  result  is  76149.     Write  this  as  units.     Then  mul- 

rv/^l  j^q  tiply  by  3  tens,  which  is  25383  tens.  Then  multiply 
by  2  hundreds,  which  is  16922  hundreds.  These  are 
called  partial  products.  Write  the  partial  products 
each  one  place  to  the  left  of  the  previous  one,  and 


25383 
16922 


$2022179      add.     The  result  is  the  total  product. 


RATIONAL  ARITHMETIC  11 

118.  To  Check  the    Work:    Let  the  multiplier  and 

multiplicand    change    places    and    proceed    as    before, 

thus : 

239 

8461 

239 

1434 

956 

1912 

2022179 

DIVISION  —  INTEGERS 

119.  Division  is  the  process  of  decreasing  a  given 
value  or  quantity  a  given  number  of  times. 

120.  Dividend  is  the  name  applied  to  the  value  or 
quantity  that  is  decreased  or  divided. 

121.  Divisor  is  the  name  applied  to  the  number  by 
which  the  dividend  is  decreased. 

122.  Quotient  is  the  number  of  times  the  divisor  is 
contained  in  the  dividend ;   the  result  of  division. 

123.  Division  is  the  reverse  of  multiplication.  Hav- 
ing the  product  and  either  of  the  factors  given,  the 
other  factor  may  be  found  by  dividing  the  product 
by  the  given  factor.  Division  may,  therefore,  be 
used  as  a  method  of  checking  multiplication,  and  vice 
versa. 


12  RATIONAL  ARITHMETIC 

ILLUSTRATED   SOLUTIONS 
124.    Problem:   1736^7=? 


248 


7  is  contained  in  17  twice,  with  3  remaining.     Write 

the  figure  2  over  (or  under)  the  7  as  the  first  figure  of  the 

7)1736      quotient.     The  3  hundreds  left  over,  combined  with  the 

Or,        next  figure,  3,  makes  33  tens.     7  is  contained  in  33  four 

7H736     times,  and  5  are  left.     Write  the  quotient  figure  4,  plac- 

^^7^     ing   it   over    (or  under)   the   3    of   the    dividend.     The 

5    tens    left  from    this    operation,   combined    with   the 

6   units,  makes  56  units.     7  is  contained  8  times  in  56.     Place 

the  8  over  (or  under)  the  6  of  the  dividend  and  the  final  quotient  is 

complete,  248. 

125.    To   Check   the   Work:    Multiply   the    quotient 
by  the  divisor.     The  result  should  be  the    dividend. 

Thus,  248X7  =  1736. 


126.    Problem  :  248963  ^  139  =  ? 

Write  the  problem  in  proper  form  for  di- 

17Q1JJL  — 

^^^^139        vision,  thus,  139)248963. 

139)248963  139  is  contained  in  248  once.     Write  1  over 

139  the  8  to  show  this.     139X1  =  139.     139  sub- 

1099  tracted  from  248  leaves  109.     To  this  annex 

the  next  figure  in  the  dividend,   9.     139   is 

contained  in  1099  seven  times.     Write  7  over 

the    9.       7  X  139  =  973.       1099  -  973  =  126. 


973 


1266 

^^^^  Bring  down  6  and  proceed  as  before,  so  con- 

153  tinning  until  all  the  figures  of  the  dividend 

139  have  been  used.     After  the  last  figure  is  used, 

I  j^  the  remainder,  if  any,  should  be  written  above 

a  line  with  the  divisor  below,  thus,  j^^- 


RATIONAL  ARITHMETIC  13 

127.  To  Check  the  Work  :  Multiply  the  whole  num- 
ber of  the  quotient,  1791,  by  the  divisor,  139,  and  add 

the  remainder,  14,  thus  : 

1791 

139 
16119 
5373 
1791 
14 

248963 

DECIMALS 

128.  All  figures  to  the  right  of  the  decimal  line  repre- 
sent parts  of  a  unit.  Each  removal  of  the  figure  one  place 
from  the  line,  to  the  right,  divides  its  value  by  ten. 
From  the  Latin  decern,  meaning  ten,  we  derivq  the  name 
decimal,  which  is  applied  to  all  values  or  quantities  repre- 
sented by  figures  written  to  the  right  of  the  decimal  line. 

From  the  fact  that  figures  representing  decimal  values  are  written 
in  exactly  the  same  way  as  to  represent  integral  vakies,  except  that 
they  appear  to  the  right  of  the  decimal  line,  the  actual  processes  of 
addition,  subtraction,  multiplication,  and  division  are  the  same  as 
for  integers. 

129.  The  only  matter  requiring  special  attention  is 
the  decimal  line  or  decimal  point. 

ADDITION   AND    SUBTRACTION  —  DECIMALS 

130.  Decimals  are  added  and  subtracted  in  exactly 
the  same  manner  as  integers.  The  essential  thing  is 
to  place  the  decimal  points  under  one  another  so  as  to 
form  the  decimal  line. 


14  RATIONAL  ARITHMETIC 

ILLUSTRATED   SOLUTIONS 

131.    Problem:     248.15  +  .43642  +  12.05  +  124.3096-f 
85.03752  +  100.075  +  2.12465=  ? 


248 

15 

43642 

12 

05 

124 

3096 

85 

03752 

100 

075 

2 

12465 

572 

18319 

After  arranging   the   figures   so  as   to   form   the 
decimal  Une,  add  as  explained  in  par.  99. 


132.    Problem:  246.75-4.88625=? 


946 
4 


241 


75000 

OQ^Q^  After   arranging  the  figures  so  as  to  form  the 

decimal  line,  subtract  as  explained  in  par.  106. 

86375 

MULTIPLICATION  —  DECIMALS 


133.  The  actual  work  of  multiplying  decimals  is 
exactly  the  same  as  for  multiplying  integers.  It  is 
necessary,  however,  to  be  able  to  locate  the  decimal 
line,  or  decimal  point,  in  the  product  with  unfailing 
accuracy. 

One  tenth  (.1)  multiplied  by  five  tenths  (.5)  equals  five  hun- 
dredths (.05).  Then  one  decimal  place  multiplied  by  one  decimal 
place  produces  tivo  decimal  places  in  the  product. 

Five  hundredths  (.05)  multiplied  by  three  tenths  (.3)  equals 
fifteen  thousandths  (.015).  That  is,  two  decimal  places  multiplied 
by  one  decimal  place  produces  three  decimal  places  in  the  product. 

From  this  we  see  that  the  product  contains  as  many  decimal 
places  as  the  multiplicand  and  multiplier  together  contain. 


RATIONAL  ARITHMETIC  15 

134.  To  Locate  the  Decimal  Point  in  the  Product  : 
Count  the  number  of  decimal  figures  in  both  factors 
together.  Place  the  decimal  line,  or  point,  that  num- 
ber of  places  from  the  right-hand  end  of  the  product. 

ILLUSTRATED   SOLUTION 

135.  Problem :   Multiply  .875  by  .63. 


.875 


Multiply  as  explained  in  par.  117,  regardless  of  the 
point. 

•^^  Since  there  are  three  decimals  in  one  factor  and  two 

'^Q^5      decimals  in  the  other,  counted  together  there  will  be  five 

5250         decimals  in  the  product.     Count  five  figures  beginning 

551*^5      with  the  right  figure  of  the  product.     Place  the  decimal 

line,  or  point,  to  the  left  of  the  fifth  figure,  thus,  .55125 

Note.  For  practice  problems  in  the  multiplication  of  decimals  see 
par.  10. 

DIVISION  —  DECIMALS 

136.  The  actual  work  of  division  of  decimals  is  the 
same  as  for  division  of  integers,  but  it  is  necessary  to 
be  able  to  place  the  decimal  point  in  the  quotient  with 
absolute  accuracy. 

Since  the  process  of  division  is  the  direct  opposite  of  that  of 
multiplication ;  since  the  dividend  of  division  is  the  product  of 
multiplication ;  since  the  quotient  of  division  is  one  of  the  factors 
of  multiplication ;  and  since  the  divisor  is  the  other  factor ;  if  we 
take  the  number  of  decimals  in  the  divisor  (one  factor)  from  the 
number  of  decimals  in  the  dividend  (the  product)  it  will  give  us  the 
number  of  decimals  in  the  quotient  (the  other  factor). 

With  this  knowledge  it  is  easy  to  understand  the  following  rule, 
which  should  be  memorized  and  carefullv  followed  under  all  circum- 
stances. 


16  RATIONAL  ARITHMETIC 

137.  To  Locate  the  Decimal  Point  in  the  Quotient  : 
Imagine  the  divisor  placed  upon  the  dividend  so  that 
the  decimal  line  of  the  divisor  covers  the  decimal  line 
of  the  dividend.  Count  the  number  of  decimal  places 
"  covered  "  in  the  dividend.  Place  the  new  decimal 
line  at  this  point,  extending  it  up  into  the  quotient. 
Proceed  as  in  ordinary  division. 

138.  To  he  absolutely  sure  that  the  decimal  line,  or 
point,  is  in  the  right  place,  it  should  be  placed  in  the 
quotient  before  any  figures  are  written  there. 

ILLUSTRATED   SOLUTIONS 

139.  Problem:   12.435^1.5=.^ 

Apply  the  above  rule  (137)  :  Imagine  1.5  placed 
upon  12.435  in  the  expression  1.5)12.435  so  that 
decimal  point  covers  decimal  point.  One  figure, 
4,  in  the  dividend  is  covered. 

Place    the    decimal  line  between   4   and  3  ex- 


8 

29 

.5)12.4 

35 

12  0 

43 

30 

135 

1 

35 

tending   it   into  quotient,  thus   1.5)12.4 
proceed  as  for  integers  (par.  126). 

Ans.  8.29. 


35     Now 


140.    Problem:   12.875 -M4  =  .^ 

|9|9_9_  Apply  the  same  rule:   Imagine  14.  placed 


1  4 


14  ')12!875  ^^    12.875    in    the    expression    14.)  12.875    so 

12  6  that  decimal  point  covers  decimal  point.    No 

decimal  figures  in  the  dividend  are  covered. 
Therefore,  the  decimal  line  remains  unchanged. 


27 
14 


^"^^  thus,  14.)  12  875     Now  proceed  as  for  integers 

M?  and  the  result  will  be  .91  g^-^^. 

^         Ans.  .919i^. 


RATIONAL  ARITHMETIC  17 

141.    Problem:   13.5 --.875=? 


Apply  the  same  rule :  Imagine  .875 
placed  on  13.5  in  the  expression  .875)13.5  so 
that  decimal  point  covers  decimal  point. 
Three  decimal  places  in  the  dividend  are 
covered.  Two  of  these  places  must  be 
visualized  with  O's.  Therefore,  the  decimal 
line  falls  between  the  third  and  fourth  places, 


15  428+ 

875)13.500  000 

8  75 

4  750 

4  375 

375  0 

350  0 

25  00 

17  50 

7  500 

7  000 

thus,    .875)13.500 


Now     proceed     as     for 


integers  and  the  result  will  be  15.428+. 


500       Ans.   15.428+. 

In  this  use  +  means  "  and  more." 

Note.     For  practice  problems  in  the  division  of  decimals  see  par.  11. 

FACTORING 

142.  Every  number  can  be  divided  by  1  and  by 
itself  without  leaving  a  remainder. 

143.  Numbers  that  can  be  divided  exactly  only  by 
themselves  and  by  1  are  Prime  Numbers.  1,  2,  3,  5, 
7,  11,  13,  17,  19,  etc.,  are  prime  numbers. 

144.  Numbers  that  can  be  divided  exactly  by  num- 
bers other  than  themselves  and  1  are  Composite 
Numbers. 

145.  Composite  numbers  are  made  up  of  two  or 
more  prime  numbers  multiplied  together.  These  are 
called  Prime  Factors.  It  is  sometimes  necessary  to 
find  what  prime  factors  go  to  make  up  a  number. 


18  RATIONAL  ARITHMETIC 

146.    Problem  :    What  are  the  prime  factors  of  420  ? 


2)420 


ILLUSTRATED   SOLUTION 

The  smallest  prime  number  contained  in  4'20 

is  2.     It  is  contained  210  times.     The  smallest 

^J^IO      prime  number  contained  in  210  is  2.      It  is  con- 

3)105      tained   105   times.      The  smallest  prime  number 

5)    35      contained  in  105  is  3.     It  is  contained  35   times. 

ry      The   smallest  prime   number  contained  in  35  is 

5.      It  is  contained  7  times.      7  is  itself    prime. 

Therefore,  the  prime  factors  of  420  are  2,  2,  3,  5,  7. 

Ans.  2,  2,  3,  5,  7. 

147.  To  Check  the  Work  :  Multiply  the  prime  factors. 
The  result  will  be  the  original  quantity,  thus,  2X2X3X 
5X7  =  420. 

LEAST    COMMON    MULTIPLE 

148.  A  Common  Multiple  of  several  numbers  is  a 
quantity  that  contains  all  of  them. 

(a)  Thus :  48  is  a  common  multiple  of  6,  8,  4,  3,  and  2,  because 
it  contains  all  of  them. 

(b)  The  product  of  the  several  numbers  is  always  a  common 
multiple  of  them.  That  is,  8X5X4  (160)  is  a  common  multiple  of 
8,  5,  and  4. 

149.  The  Least  Common  Multiple  of  several  numbers 
is  the  least  quantity  that  contains  all  of  them. 

{a)  Thus  :  24  is  the  least  common  multiple  of  6,  8,  4,  3,  and  2, 
because  it  is  the  least  number  that  contains  all  of  them. 

(b)  The  product  of  the  prime  factors  of  several  numbers  is  the 
least  common  multiple  of  those  numbers. 


2)24 

15 

36 

9 

2)12 

15 

18 

9 

3)  6 

15 

9 

9 

3)  2 

5 

3 

3 

RATIONAL  ARITHMETIC  19 

150.  To  Find  the  Least  Common  Multiple  of  Several 
Numbers  :  Find  the  prime  factors  of  the  numbers. 
Multiply  these  factors  together.  The  result  will  be 
the  least  common  multiple  of  the  original  numbers. 

ILLUSTRATED   SOLUTION 

151.  Problem  :  Find  the  least  common  multiple  of 
24,  15,  36,  and  9. 

Write  the  numbers  in  line.  2  is 
the  smallest  prime  factor  contained 
in  anv  two  or  more.  Divide  bv  2 
where  possible,  bringing  down  num- 
bers of  which  2  is  not  a  factor. 
^^11  The  result  is  12,  15,  18,  9.     2  is  a 

2X2X3X3X2X5  =  360.         factor  of   12  and   18.     Proceed  as 

before.  The  result  is  6,  15,.  9,  9. 
3  is  a  prime  factor  of  15  and  9.  Proceed  as  before.  The  result  of 
this  is  2,  5,  3,  3.  3  is  a  factor  of  3  and  3.  Proceed  as  before. 
The  result  is  2,  5,  1,  1,  each  of  which  is  a  prime  number.  The 
prime  factors  then  are  2,  2,  3,  3,  2,  5.  Multiply  these  and  we  ob- 
tain 360,  which  is  the  least  common  multiple. 

152.  To  Check  the  Work  :  Divide  the  least  common 
multiple  by  each  of  the  original  numbers,  thus : 

360-=-24  =  15  360^15  =  24 

360-r-36  =  10  360^   9  =  40 

GREATEST   COMMON   DIVISOR 

153.  A  Common  Divisor  of  two  or  more  numbers  is 
any  number  that  will  exactly  divide  each  of  them. 


20  RATIONAL  ARITHMETIC 

154.  The  Greatest  Common  Divisor  is  the  greatest 
number  that  will  exactly  divide  each  of  them. 

7  is  a  common  divdsor  of  21,  35,  and  14.  It  will  divide  each  of 
them  exactly.  Some  numbers  have  no  common  divisor.  No  num- 
ber will  divide  both  9  and  8  exactly.  Such  numbers  are  said  to  be 
prime  to  each  other. 

155.  To  Find  the  Greatest  Common  Divisor  :  Divide 
the  greater  number  by  the  lesser.  Then  divide  the 
first  divisor  by  the  remainder,  and  so  continue  until 
the  division  is  exact,  or  there  is  a  remainder  of  one. 
If  exact,  the  last  divisor  is  the  greatest  connnon  divisor. 

If  the  remainder  is  one,  there  is  no  common  divisor. 
The  numbers  are  prime  to  each  other. 

ILLUSTRATED   SOLUTIONS 

156.  Problem  :  What  is  the  greatest  common  divisor 
of  351  and  459? 


1 

351)459 

351 

3 

108)351 

324 

4 

27)108 

108 

Divide  459  by  351  (par.  126).  The  re- 
mainder is  108.  Divide  351  by  108  and  the 
remainder  is  27.  Divide  108  by  27.  There 
is  no  remainder.  Therefore,  27  is  the 
G.  C.  D.  of  459  and  351. 


G.  CD.  =  27. 


157.    To  Check  the  Work  : 

459-^27  =  17  351^27  =  13 


RATIONAL  ARITHMETIC  21 

158.    Problem  :    Find  the  G.  C.  D.  of  209,  247,  and 
456. 

1 


209)247 

209       5 

38)209 
190    2 

19)38 
38 

24 


Find  the  G.  C.  D.  of  209  and  247,  as  in 
the  previous  problem.     Result,  19. 


19)456 

^^  Find  the  G.  C.  D.  of  19  and  456,  which  is 

76  19.     Therefore,  19  is  the  G.  C.  D.  of  all. 

76 

0 
G.  C.  D.  =  19. 

159.    Problem  :   Find  the  G.  C.  D.  of  943  and  35. 
26 


35)943 
70 

243 
210 

1 

33)35 
33  16 

2)33 

2 

13 
12 

1 

No  G. 

CD. 

Proceeding  as  above,  we  derive  a  final  re- 
mainder of  1.  Therefore,  the  numbers  are 
prime  to  each  other. 


FRACTIONS 

160.  A  Fraction  is  a  part  of  an  integer. 

(a)  Any  quantity  or  value  may  be  divided  into  any  number  of 
equal  parts.     One  or  more  of  these  equal  parts  constitutes  a  fraction. 

(b)  If  a  quantity  be  divided  into  four  equal  parts,  each  of  them 
will  be  known  as  one-fourth  {^)  and  three  of  them  would  be  three- 
fourths  (f ).  If  the  whole  were  divided  into  a  thousand  equal  parts, 
each  part  would  be  called  one  one-thousandth  (toVo")  ^^^  ^^^.V  num- 
ber of  these  parts  could  be  made  to  form  the  fractional  value,  as 
748  thousandths  (1^*7^). 

161.  A  fraction  is  composed  of  two  terms.  One 
term  shows  the  number  of  parts  into  which  the  original 
integer  was  divided.  The  other  term  shows  the  num- 
ber of  these  equal  parts  taken  to  make  the  fraction. 

162.  Denominator  is  the  name  applied  to  the  term 
showing  into  hoiv  many  parts  the  integer  was  divided. 

163.  Numerator  is  the  name  applied  to  the  term 
showing  the  number  of  these  equal  parts  used  in  the 
fraction. 

164.  There  are  two  methods  of  fractional  notation, 
common  fractions  and  decimal  fractions. 

165.  A  Common  Fraction  is  expressed  by  writing 
the  numerator  above  the  denominator  with  a  line 
between,  thus,  f . 

22 


RATIONAL  ARITHMETIC  23 

166.  A  Decimal  Fraction  is  any  fraction  whose 
denominator  is  10  or  any  multiple  of  10. 

It  is  not  necessary  to  write  the  denominators  of  decimal  fractions. 
All  that  is  required  is  to  write  the  numerator  and  place  it  in  the 
proper  decimal  column  to  show  what  denominator  is  intended,  thus^ 
six-tenths  (y^)  may  be  written  .G.  Six-thousandths  may  be  ex- 
pressed by  placing  the  figure  six  in  the  proper  decimal  column,  thus, 
.006,  and  so  on. 

167.  Every  fractional  value  may  be  expressed  either 
as  a  common  fraction  or  as  a  decimal  fraction. 

In  some  cases  it  is  easier  to  use  the  common  fraction  form ; 
while  in  others  it  is  easier  to  use  the  decimal  form  expressing  the 
same  value. 

168.  A  Proper  Fraction  is  a  common  fraction  whose 
numerator  is  smaller  than  its  denominator.  Its  value 
is  less  than  one. 

169.  An  Improper  Fraction  is  a  common  fraction 
whose  numerator  is  larger  than  its  denominator.  Its 
value  is  more  than  one. 

170.  A  Mixed  Number  is  an  integral  quantity  and 
a  fraction  taken  together,  as,  1^. 

171.  In  its  decimal  form  this  quantity  would  be 
called,  a  Mixed  Decimal  and  would  be  written  1.25. 

172.  It  is  not  always  possible  to  reduce  a  fractional 
value  to  an  exact  decimal.  In  such  cases  it  is  neces- 
sary to  retain  a  common  fraction  as  part  of  a  decimal, 
thus,  .16f,  read  sixteen  and  two-thirds  hundredths. 

173.  A  Complex  Decimal  is  a  decimal  whose  right- 
hand  place  is  a  common  fraction. 


24  RATIONAL  ARITHMETIC^ 

J 

174.   A  Complex  Fraction  is  a  common  fraction,  the 
right-hand  place  of  whose  numerator  is  itself  a  fraction. 


thus, 


161 
100 


175.  Every  common  fraction  represents  an  unper- 
formed division. 

f  is  the  result  of  performing  the  operation  8^9.  It  may  be  read 
"  eight  divided  by  9  "  or  it  may  be  read  "  eight-ninths."  |-  and  also 
2-i-3  may  each  be  read  "2  divided  by  3." 

176.  Reduction  of  Fractions  is  the  process  of  chang- 
ing the  form  of  a  fraction  without  changing  its  value. 

177.  Reduction  of  fractions  comprises  : 

Changing  to  lower  terms. 

Changing  to  higher  terms. 

Changing  improper  fractions  to  mixed  numbers. 

Changing  mixed  numbers  to  improper  fractions. 

Changing  decimals  to  common  fractions. 

Changing  common  fractions  to  decimals. 

CHANGE   TO   LOWER   TERMS 

178.  Common  fractions  are  in  their  lowest  terms 
when  the  numerator  and  denominator  are  prime  to 
each  other ;  that  is,  when  they  have  no  common 
factors,  no  common  divisor. 

179.  Decimal  fractions,  when  written  in  the  form 
of  common  fractions,  may  be  reduced  to  lower  terms 
when  their  numerator  and  denominator  are  not  prime 
to  each  other. 


RATIONAL  ARITHMETIC 


15 


The  decimal  fraction,  four-tenths,  if  written  in  its  common  frac- 
tion form  ^,  shows  at  once  that  its  real  or  lowest  value  is  f  because 
2  is  a  common  factor  of  both  4  and  10.  In  other  words,  if  an  integer 
is  divided  into  ten  parts  and  four  of  these  parts  are  taken,  the  value 
of  the  fraction  would  be  exactly  the  same  as  if  the  integer  had  been 
divided  into  five  parts  and  two  of  these  parts  taken. 

180.  To  Change  to  Loiver  Terms  :  Strike  out  all 
factors  common  to  both  the  numerator  and  the  de- 
nominator; or,  divide  both  the  numerator  and  the 
denominator  by  their  G.  C.  D. 


ILLUSTRATED   SOLUTIONS 
181.    Problem :  Reduce  414  to  lowest  terms. 


4 

^^ 

5 

Ans. 

Or, 


5' 


Seven  is  a  factor  of  both  420  and  o'l5.  Dividing 
the  numerator,  420,  by  7  gives  a  new  numerator  of 
60.  Dividing  the  denominator,  52o,  by  7  gives  a 
new  denominator  of  75.  Five  is  a  common  factor 
of  both  60  and  75.  Dividing  60  by  5  gives  a  new 
numerator  of  12.  Dividing  75  by  5  gives  a  new 
denominator  of  15.  Three  is  a  common  factor  of 
12  and  15.  Divide  12  by  3  and  we  have  a  new 
numerator  of  4.  Dividing  15  by  3  gives  a  new 
denominator  of  5.  No  factor  is  common  to  4  and  5. 
Therefore,  4  is  the  lowest  equivalent  of  ^^ 


420)525 
420 


105)525 
525 

4 


105)420^4 
105)525  5 


Find  the  G.  C.  D.  of  420  and  525  (pars.  155 
and  156).  The  G.  C.  D.  is  105.  420^-105  =  4. 
Therefore,  the  new  numerator  is  4.  525  -M05  =  5. 
Therefore,  the  new  denominator  is  5,  making  the 


fraction  ^. 


26  RATIONAL  ARITHMETIC 


1- 
182.    Problem :    Chanaje  -^  to  lowest  terms. 

^     9 


1^ 


n_ 

14 

8 

40 

7 

M_ 

7 

4rar 

20 

20 

v4n5. 

7 

2  n 

This  is  a  complex  fraction.     The  first  step  is  to 

J^  =z  ^  change  the  numerator  and  the  denominator  to  the 

9        27  same  kind   of  parts,   thirds.     One  unit  =  f ;  then 

Ans.    2T'       lf~f '  ^  units  =  -^3^.     Then  the  numerator  equals 

5  thirds  and   the  denominator  equals  27  thirds ; 

■5  and  27  being  prime  to  each  other,  -^^  is  in  its  lowest  terms. 

2- 
183.    Problem :    Chanaje  —  to  lowest  terms. 

^     8 


Proceeding  as  before,  we  find  that  the  complex 
fraction  is  equal  to  ^.  Both  terms  may  be  divided 
by  2,  producing  -^q'  ;  7  and  20  being  prime  to  each 
other,  -^Q  is  in  its  lowest  terms. 


Note.  For  practice  problems  in  reducing  fractions  to  lowest  terms 
see  par.  12. 

CHANGING   TO   HIGHER   TERMS 

184.  In  handling  common  fractions  in  business 
problems,  it  is  sometimes  necessary  to  change  a  com- 
mon fraction  from  its  lowest  terms  to  an  equivalent 
common  fraction  of  some  higher  denomination. 

To  do  this  it  is  necessary  to  introduce  such  factors 
into  both  the  denominator  and  numerator  as  will 
change  the  value  of  the  denominator  from  the  given 
figure  to  that  required. 

185.  To  Change  a  Common  Fraction  to  a  Required 
Denominator:    Divide    the    required   denominator   by 


RATIONAL  ARITHMETIC  27 

the  given  denominator.  Multiply  both  the  numerator 
and  the  denominator  of  the  given  fraction  by  the 
quotient  thus  obtained. 

ILLUSTRATED   SOLUTION 

186.  Problem  :   Change  f  to  525ths. 

105 

5^5^5  Five  is  contained  105  times  in  5'i5;   therefore, 

5  must  be  multiplied   hv   105   to  produce  515. 

4X105  =  420      Multiplying   4   by    105   gives   420   for   the   new 

^VlO^  —  ^9^      numerator.     Multiplying   5    by    105    gives   525, 

.  ^20         ^^^  required  denominator. 

./j-ito.      .525* 

Note.     For  practice  problems  in  changing  fractions  to  higher  terms  see 
par.  13. 

CHANGING   AN   IMPROPER   FRACTION   TO   A   MIXED 

NUMBER 

187.  The  denominator  shows  into  how  many  parts 
the  original  integer  has  been  divided.  If  the  number 
of  parts  taken  is  greater  than  the  number  into  which 
the  integer  was  divided,  as  is  the  case  with  every  im- 
proper fraction,  then  the  value  of  every  improper 
fraction  must  be  greater  than  the  value  of  the  original 
integer.  It  will  be  as  many  times  the  original  integer 
as  the  denominator  is  contained  in  the  numerator. 
If  the  denominator  is  not  contained  in  the  numerator 
an  even  number  of  times,  the  remaining  number  of 
parts  would  constitute  the  number  of  fractional  units 
that  are  left. 


28  RATIONAL  ARITHMETIC 

188.  To  Change  an  Improper  Fraction  to  a  Mixed 
Number :   Divide  the  numerator  by  the  denominator. 

ILLUSTRATED   SOLUTION 

189.  Problem  :   Change  -^-  to  a  mixed  number. 

3-1 


9)32 
27 
5 
Ans.  34 


The  denominator,  9,  is  contained  3f  times  in 
the  numerator,  32.     Therefore,  %^=3^. 


9' 


Note.  For  practice  problems  in  changing  improper  fractions  to  mixed 
numbers  see  par.  14. 

CHANGING   A   MIXED   NUMBER   TO   AN   IMPROPER 

FRACTION 

190.  The  denominator  of  a  fraction  shows  the  num- 
ber of  parts  into  which  the  unit  has  been  divided. 
Every  unit  of  a  mixed  number  then  will  contain  as 
many  of  these  parts  as  are  expressed  by  the  denomina- 
tor. The  fractional  value  of  the  whole  then  may  be 
found  by  multiplying  the  integral  number  by  the 
denominator  of  the  fraction  and  adding  the  numerator 
of  the  fraction  to  this  result. 

191.  To  Change  a  Mixed  Number  to  an  Improper 
Fraction:  Multiply  the  integer  by  the  denominator 
of  the  fraction.  To  the  result  add  the  numerator  of 
the  fraction.  The  total  should  be  written  as  the 
numerator  of  the  new  fraction.  The  denominator 
remains  unchanged. 


RATIONAL  ARITHMETIC  29 

ILLUSTRATED   SOLUTION 

192.  Problem  :   Change  3|  to  an  improper  fraction. 

^y\c>  =  ^^  The  value  of  three  units  is  twenty-four  eighths. 

24-(-5  =  29         Twenty-four    eighths    pUis    five    eighths     equals 
j4_flS,   ^,        twenty-nine  eighths.     Therefore,  3f  equals  -^^. 

Note.  For  practice  problems  in  changing  mixed  numbers  to  improper 
fractions  see  par.  15. 

CHANGING   A   DECIMAL   FRACTION   TO   A   COMMON 

FRACTION 

193.  To  Change  a  Decimal  Fraction  to  a  Common 
Fraction:  Write  the  decimal  in  its  common  fraction 
form  and  then  reduce  to  lowest  terms. 

ILLUSTRATED   SOLUTIONS 

194.  Problem:   Change  .125  to  a  common  fraction. 

1  Write  •I'^o  in  its  fraction  form.     It  is  apparent 

■^^<^_  t  that  1*25  is  a  common  divisor  of  both  the  numerator 

iOOO"     8  and   the  denominator.     125   is   contained   once   in 

8  the  numerator.     U25  is  contained  eight  times  in  the 

A.ns.   ^.  denominator. 

195.  Problem:   Change  .16f  to  a  common  fraction. 

Writing  .16§  as  a  common  fraction  produces  a 
complex  fraction.  Reduce  this  complex  fraction  to 
its  lowest  terms  (par.  IS'i).  The  numerator  con- 
tains fifty  thirds,  the  denominator  contains  three 
hundred  thirds,  making  -^^-q  or  ^. 


Note.     For  practice  problems  in  changing  decimal  fractions  to  common 
fractions  see  par.  16. 


100  = 

=  300 

1 
m  _ 

_1 

%m  6 

6 

Ans.  -5- 

30  RATIONAL  ARITHMETIC 

CHANGING   A   COMMON   FRACTION   TO   A   DECIMAL 

FRACTION 

196.  Every  common  fraction  is  the  statement  of 
an  unperformed  division  (par.  175).  The  result  of 
performing  this  division  is  a  decimal. 

(a)  The  principle  involved  in  changing  a  common  fraction  to  a 
decimal  is  practically  the  same  as  that  for  changing  an  improper 
fraction  to  a  mixed  number. 

(6)  In  changing  from  an  improper  fraction  to  a  mixed  number  all 
the  work  is  on  the  integral,  or  left,  side  of  the  decimal  line ;  while  in 
changing  from  a  common  fraction  to  a  decimal  fraction  the  work  is 
all  to  the  right  of  the  decimal  line. 

(c)  In  changing  from  mixed  numbers  to  mixed  decimals,  the 
work  is  on  both  sides  of  the  line. 


197.  To  Change  a  Common  Fraction  to  a  Decimal 
Fraction  :   Divide  the  numerator  by  the  denominator. 

Decimal  values  are  seldom  carried  beyond  the  sixth  decimal  place. 
Any  fraction  remaining  at  this  point  is  usually  disregarded,  although 
when  absolute  accuracy  is  desired  the  decimal  should  be  carried  out 
until  exact,  or  the  fraction  should  be  retained,  making  a  complex 
decimal. 

ILLUSTRATED   SOLUTIONS 

198.  Problem  :    Change  i  to  a  decimal  fraction. 

1 125  Divide   the   numerator   by   the    denominator, 

8)1 1 000  fi^^*  placing  the  decimal  point  in    the  quotient, 

Ans      125       ^^  shown  in  pars.  139,   140,  141.     Three  decimal 
places  will  be  used,  making  the  result  .l!25. 


RATIONAL  ARITHMETIC  31 

199.  Problem  :  What  is  the  decimal  value  of  {i  ? 

lftRR2  Solve  this  problem  in  the  same  way  as  the 
previous  one.     After  two  decimal  places  have 

1  fir  \  1  o  f\f\f\ 

ID)  lo  uuu  been   used,    we   find   that   the   remainder   will 

i-^  ^  continue    to    repeat   itself.     This    shows    that 

1  00  the  division  will  never  be  exact.     No  matter 

90  how  far  carried,  the  decimal  figure  will  be  6. 

Tq  We  may,  therefore,  stop  at  any  decimal  place, 

J  ftf»fi2  retain  the  fraction  ^,  and  make  the  result  a 

^*  complex  decimal,  .866|^,  .8666|^,  or  .86§,  etc. 

200.  Problem  :   Change  2^  to  its  decimal  form. 

P  Reducing  the  common  fraction  ^  as  shown  above, 

5)4 10  the  result  is  .8.     The  integer  2  remains  the  same 

Ans.   2.8.    ^^^  t^^^  result  is,  therefore,  2.8. 

Or, 


^5  —    5 

2|8  Changing  2f  to  fifths,  we  have  ^,  which,  when 

5)1410  reduced  according  to  rule,  gives  2.8. 

Ans.  2.8. 

Note.  For  practice  problems  in  changing  common  fractions  to  decimal 
fractions  see  par.  17. 

ADDITION    OF   FRACTIONS 

201.  Only  numbers  representing  like  values  or  like 
quantities  or  like  parts  of  such  values  and  quantities 
can  be  added  (par.  96,  a,  b). 

(a)  In  adding  decimals  all  that  is  necessary  is  to  carefully  arrange 
the  numbers  so  that  the  decimal  points  form  a  decimal  line.     In 


32  RATIONAL  ARITHMETIC 

this  way  tenths  come  over  tenths,  hundredths  over  hundredths,  and 
so  on,  thus  making  it  possible  to  add  Uke  parts. 

(6)  In  adding  common  fractions  it  is  necessary  to  change  all  frac- 
tions to  equivalent  fractions  having  the  same  denominator. 

202.  To  Add  Common  Fractions  :  Find  the  L.  C.  M. 
of  all  the  denominators  (par.  150).  Use  this  L.  C.  M. 
as  a  common  denominator.  Change  each  given  frac- 
tion to  a  fraction  having  this  denominator  (par.  185). 
Add  the  numerators  of  the  new  fractions.  The  result 
is  the  numerator  of  the  sum.  The  L.  C.  M.  of  the 
given  denominators  is  the  denominator  of  the  sum. 
Reduce  the  sum-fraction  to  its  lowest  terms. 

ILLUSTRATED   SOLUTIONS 

203.  Problem:  f+f+TV+l=  ? 

2)3-8-12-6 

2)3-4-   6-3 

3)3_2—    3  —  3  We  find  the  L.  C.  M.  of  the  given  de- 

1  _Q \  —\  nominators,  3,  8,  1  "2,  6,  to  be  24  (pars.  150, 

/-.     /-»     r.    ./-»      r»j  5k         151).     Change  each  of  the  given  fractions 

^X^X^X^      ^^  to  twenty-fourths,    f  =  if ,  f  =  M' T2  =M 

--16  andf  =  |f. 

3  As  the  denominators  are  all  the  same 

5         ^  and  we  are  to  add  only  the  numerators, 

8  it  will  save  time  and  confusion  if  we  simply 

7  write  the  numerators  16,  15,  14,  20. 

]^2  Added,  we  have  fl^,  which  reduced  to  a 

K  mixed  number  (pars.  188,  189)  equals  2^x« 


65 

24 
*  This  L.  C.  M.  can  be  determined  mentally. 


a,=^ii      Ans.  m 

24 


RATIONAL  ARITHMETIC  33 

204.  Problem:   12t+9i+23f+19f  =  ? 

2)5-2-8-6 
2)5-1-4-3 

5  —  1  —  2  —  3  Arrange   the   mixed   numbers  in   a 

2X2X5X2X3  =  120  column.     Add    the    fractions    as    ex- 

-JQ4. qrj  plained  in  the  previous  solution.     The 

^^  result  will  be  3  0j.  =  £>_6_i_ 

qj_ r^r\  lesLUL   will   ue    J  20       ^12  0' 

^  The  Y2V  ^i^^  ^^  t^^  fraction  of  the 

238  —    45  gj^j^l  gyj^^ 

19f — 100  Add  the  2  with  the  given  integers. 

2        tI i  =  2 1^  The  total  sum  is  Q5^\. 

^^1  2  0 

./xTlS .    \)D  -^  2  0  • 

205.  Problem:   12.8  +  19i+14.875+5.8i=  ? 

12.8  Mixed  numbers,  mixed  decimals  and  com- 

jg  ^  plex  decimals  can  be  added,  but  it  is  neces- 

sary first  to  change  the  mixed  numbers  to 
mixed  decimals  ;  that  is,  19|  must  be  changed 
to  19.5.     The  complex  decimals  must  be  re- 


14.875 

5.833^ 


oJ.UUo3^  duced    to    the    same   order   as    the    longest 

Ans.   53.008^.      decimal;  that  is,  5.8^  must  be  changed  to 

5.833^.     x\fter  these  reductions  have  been 
made,  add  as  explained  in  the  previous  solutions. 

Note.     For  practice  problems  in  addition  of  fractions  see  par.  18. 

SUBTRACTION    OF   FRACTIONS 

206.    Subtraction  of  fractional  values  is  performed 
in  the  same  way  as  subtraction  of  integral  quantities. 

As  in  addition,  it  is  necessary  to  change  the  given  fractions  to 
fractions  having  a  common  denominator. 


34  RATIONAL  ARITHMETIC 

ILLUSTRATED   SOLUTION 
207.    Problem:  412i-204f  =  ? 

The  given  denominators  are  9  and  2.     The 

^        ^'  common   denominator  is   18.     ^  =  ^8^5  l^~T§- 

41/2^      p  Ten  cannot  be  taken  from  9.     Take  one  from 

204|-      10  the    units    cohimn    of    the    integers.      l=Tf- 

rtf\iy  iT  _9    -|-J_8___2  7  27_X0_17  ryyi  rlifff^rpnp*^ 

207         T¥  18^  I  r8~T8-      T¥     T8^  — T¥-      ^'^^    omercnce 

between  the  integers  is  207.     The  total  dif- 


Ans.  207ii. 

Note.     For  practice  problems  in  subtraction  of  fractions  see  par.  19. 


ference  is  207^1^. 


MULTIPLICATION    OF   FRACTIONS 

208.  When  an  integer  is  multiplied  by  an  integer, 
the  product  shows  a  value  greater  than  either. 

209.  When  an  integer  is  multiplied  by  a  fraction, 
the  product  shows  a  value  less  than  the  integer. 

(a)  5X5  =  ^^5.  Both  factors  are  integral  numbers  and  the  pro- 
duct, 25,  shows  an  increase  in  valuCo 

(6)  ^X2  =  f.  f  is  a  fraction  and  the  product  shows  a  value  less 
than  2. 

(c)  The  expression  ^X2  may  be  read  ^  of  2.  |^X2,  and  ^  of  2 
represent  the  same  arithmetical  operation. 

(d)  The  sign  "X"  and  the  word  "of"  are  interchangeable  in 
fractional  computation. 

210.  To  Multiply  a  Fraction  by  a  Fraction  :  Multiply 
the  numerators.  The  result  will  be  the  numerator  of 
the  product.  Multiply  the  denominators.  The  result 
will  be  the  denominator  of  the  product.  Reduce  the 
new  fraction  to  its  lowest  terms. 

The  work  may  be  simplified  by  first  casting  out  factors  that  may 
be  common  to  one  of  the  numerators  and  one  of  the  denominators. 


RATIONAL  ARITHMETIC  35 

ILLUSTRATED   SOLUTIONS 

211.  Prohlem  :   Multiply  f  by  |. 

Multiply    the    numerators,    2X4  =  8.     This    is 
%yi^  =  ^j  the  numerator  of  the  product.     Multiply  the  de- 

Ans.   #y.       nominators,  3X9  =  27.     This  is  the  denominator 
of  the  product. 

212.  Problem:  f  of|ofT'o=? 

-  of  -  of  —  =  —  Multiply  the  numerators,  5X2X7  =  70. 

o        o         lu      /*-*u  This  is  the  numerator  of  the  new  fraction. 

70  ^^  8X3X10  =  240.     This  is  the  denominator 

24^      '24  of  the  new  fraction.     2^*0  reduced  to  the 

J 'no    JL-  lowest  terms  equals  ^V- 


Or, 

1         1  The  solution  may  be  simplified  by  cast- 

^     f  ^     £    7  _  7  ing  out  factors  as  follows :     Five  is  con- 

8        3        1(^     24         tained  in  the  numerator  of  the  fraction  |^ 

of  and  in  the  denominator  of  the  fraction  yo  . 

\  Casting  out  makes  the  first  fraction  ^  and 

J  >7  9     -J-  the  last  ^.     Two  is  a  common  factor  of  the 

'  numerator  of  f  and  the  denominator  of  ^ 

and    may    be    cast    out    of    each.     Then 

multiply  1X1X7  =  7  for  the  numerator  of  the  product.      8 X3  =  24 

for  the  denominator  of  the  product. 


213.  To  Multiply  an  Integer  by  a  Fraction  :  Multiply 
the  integer  by  the  numerator  and  divide  the  product 
thus  obtained  by  the  denominator  of  the  fraction. 


36  RATIONAL  ARITHMETIC 

ILLUSTRATED  SOLUTION 

214.  Problem  :  Multiply  324  by  -. 

5 

324 
4 

5  Applying  the  above  rule,  4  X324  =  1296. 

5)1296  *  1296-^5  =  259i. 

259i 

Ans.  259i. 

215.  To  Multiply  Mixed    Numbers  :    Multiply  the 
fractions  ;    multiply  the  integers  ;   combine  the  results. 

ILLUSTRATED   SOLUTIONS 

216.  Problem  :  Multiply  246  by  23^. 


246  1964 


First  multiply  the  integer  by  the  fraction. 


23-  5)984                                         "x.vfe^x  ^j  ...V.  x.«.^..xwxx. 

'                The  product  of  246  by  the  numerator,  4,  is 

1^"5  984,  divided  by  5  gives  196f  as  the  partial 

738  product  by  \. 

492  Multiply  246  by  3  and  by  2  as  explained  in 

5854^  P^^*   ^^^'     ^^^  the  results.     The  answer  is 

Ans.  5854t.      ^^''^^• 

217.    Problem  :  Multiply  68f  by  24.     . 

Multiply  as  in  the  previous  problem.  Multi- 
plying the  integer  24  by  the  numerator,  2,  of 
the  fraction  f ,  gives  48.  Dividing  this  by  3 
gives  16.  This  is  the  partial  product  by  f. 
Multiply  68  by  24  (par.  117).  Add,  and  the 
1648  result  is  1648. 

Ans,  1648. 


68f 

16 

24 

3)48 

16 

272 

136 

RATIONAL  ARITH^ylETIC  37 


218.    Problem  :   Multiply  243|  by  42 


2431 

2k/  i  —     8 
3  A  5  "~  15 

42t 

5)972 

8 
1  5 

8 

1941 

1941 

6 

3)84 

28 

28 

486 

972 

10428 

1  4 
1  5 

Multiply  the  fraction  of  the 
multiplicand  by  the  fraction  of 
the  multiplier.  The  product  of 
the  numerators,  4  and  2,  equals  8. 
The  product  of  the  denominators, 
5  and  3,  equals  15.  The  first 
partial  product  is  j-^. 

Multiply   the  integral  part  of 
the  multiplicand  by  the  fraction 
of    the    multiplier.     Four    times 
243   equals  972   (par.   214).     972 
Ans.    10428|i.  divided  by  5  equals  194f,  which 

is  the  second  partial  product. 
Multiply  the  fractional  part  of  the  multiplicand  by  the  integral 
part  of  the  multiplier  (par.  214).     42X2  equals  84.     84  divided 
by  3  equals  28,  which  is  the  third  partial  product. 

Multiply  the  integer  of  the  multiplicand  by  the  integer  of  the 
multiplier  (par.  117).  Add  the  partial  products  and  the  total  is 
10428if. 

Note.     For  practice  problems  in  multiplication  of  fractions  see  par.  20. 

DIVISION   OF   FRACTIONS 

219.  Division  of  fractions,  like  division  of  integers, 
is  the  direct  opposite  of  multiplication, 

220.  When  an  integer  is  divided  by  an  integer,  the 
result  shows  a  decrease.  When  an  integer  is  divided 
by  a  fraction  the  result  shows  an  increase. 

Ten  divided  by  5  equals  2.  That  is,  one  integer 
divided  by  another  gives  a  decrease.  Ten  divided 
by  one-half  equals  20,  for  in  10  units  there  are  20 
halves. 


38  RATIONAL  ARITHMETIC 

221.  To  Divide  Fractions  by  Fractions  :  Multiply  the 
numerator  of  the  dividend  by  the  denominator  of  the 
divisor.  The  result  is  the  numerator  of  the  quotient. 
Multiply  the  denominator  of  the  dividend  by  the 
numerator  of  the  divisor.  The  result  is  the  denomina- 
tor of  the  quotient.  Reduce  the  quotient  to  its  lowest 
terms. 

Or :  Invert  the  divisor  and  proceed  as  in  multi- 
plication. 

ILLUSTRATED   SOLUTIONS 

222.  Problem  :  Divide  t  by  f. 

-|-i-f  =  ij=li  The  numerator  of  the  dividend  is  4  and  the 

J  -ii  denominator   of   the   divisor    is   3.     4X3  =  12, 

^'  which  is  the  numerator  of  the  quotient.     The 

denominator  of  the  dividend  is  5  and  the  numerator  of  the  divisor 
is  2.  2X5  =  10.  The  denominator  of  the  quotient  is  10.  f|  is 
an  improper  fraction,  reduced  to  a  mixed  number  equals  l^^. 

Or, 

t"^f  =  ?  In  the  problem  f  ^f,  the  dividend  is  f  and 

tX f  =  i^  =  li       the  divisor  f .     Inverting  f  gives  f .     4X3  =  12. 

223.  To  Divide  Mixed  Numbers  :  Change  both  the 
divisor  and  the  dividend  to  improper  fractions  having 
a  common  denominator.  Divide  the  numerator  of  the 
dividend  by  the  numerator  of  the  divisor.  The  result 
will  be  the  quotient  required. 


RATIONAL  ARITHMETIC 


39 


ILLUSTRATED   SOLUTIONS 


224.    Problem  :  Divide  $124.50  by  23f. 


5  26 

71)373  50 
355 

18  5 
14  2 

4  30 
4  26 

In  tliis  problem,  both  the  dividend  and 
the  divisor  must  be  changed  to  thirds. 

$124.50  should  be  changed  to  thirds  by 
multiplying  by  three,  which  gives  373.50 
thirds. 

23f  =  ^  (pars.  191,  192).  Divide  373.50 
by  71  (par.  140).     The  result  is  $5.26y*Y- 


A 

71 


Ans.  $5.26-^. 


Problem  :  Divide  248^  by  34. 


n'2  7 


85 


170)1244 
1190 


54     27 


Changing  the  dividend,  2484,  to  fifths  equals 
1244  fifths.  Changing  the  divisor,  34,  to  fifths 
equals  170  fifths.  Divide  the  numerator  of  the 
dividend,  1244,  by  the  numerator  of  the  divisor, 
170.     The  result  is  7||. 


170     85 
225.    Problem  :   Divide  433^  by  18f . 


23 


112)2601 

224 
361 
336 
25 


Change  the  fraction  in  the  dividend  and 
the  fraction  in  the  divisor  to  fractions  having 
a  common  denominator.  433^  =  433|^  and 
18f  =18|.  Then  433i-M8f  is  the  same  as 
433f  divided  by  18|^.  Changing  the  divi- 
dend to  sixths,  we  have  2601  sixths.  Chang- 
ing the  divisor  to  sixths,  we  have  112  sixths. 

25 


Ans.  23i^.    112      Dividing  2601  by  112,  gives  23^ 

Note.     For  practice  problems  in  division  of  fractions  see  par.  21. 
For  practice  problems  involving  the  use  of  fractions  and  decimals  see 
par.  22. 


DENOMINATE   NUMBERS 

226.  A  Denominate  Number  is  a  number  expressed 
in  units  of  weight,  measure,  or  value. 

227.  A  Simple  Denominate  Number  is  a  quantity 
expressed  in  a  single  denomination. 

4  pounds,  5  bushels,  2  quarts  are  simple  denominate  numbers. 

228.  A  Compound  Denominate  Number  (usually 
called  a  compound  number)  is  a  quantity  expressed 
in  two  or  more  different  denominations. 

(a)  1  year,  9  months,  and  9  days ;  3  pounds,  9  shillings,  and  4 
pence  are  compound  numbers. 

(6)  Tables  of  weight,  measure,  distance,  values,  etc.,  will  be  found 
on  pages  130  to  146. 

(c)  In  the  follow^ing  illustrated  solutions,  English  money  is  used 
throughout  for  the  purpose  of  uniformity.  The  solutions  are  equally 
applicable  to  any  or  all  the  tables. 

REDUCTION    OF 
DENOMINATE   NUMBERS 

ILLUSTRATED   SOLUTION 

229.  To  Reduce  a  Compound  Denominate  Number 
io  a  Simple  Denominate   Number  of  Equivalent  Value: 

40 


RATIONAL  ARITHMETIC 


41 


Problem  :  Reduce  £5  Ss  9d  to  pence. 


5 

100 

8 

108 

1296 

9 

1305 


=  £ 


=  s 


=  d 


J. 


ins.  ISOBd. 


Since  there  are  205  in  £l,  in  £5  there  are  5 
times  20  or  IOO5.  £0  Ss  equals  lOSs.  Since 
there  are  12(Z  in  Is,  in  IO85  there  are  12  times 
108,  which  is  1296J.  If  there  are  UQQd  in 
£5  8s,  in  £5  8s  9d  there  are  lS05d. 


Note.     For  practice  problems  in  reducing  compound  denominate  num- 
bers to  simple  denominate  numbers  of  equivalent  value  see  par.  23. 


ILLUSTRATED   SOLUTION 


230.    To  Change  to   Higher  Denomination: 


Problem  :    Reduce  1305  pence  to  higher  denomina- 
tion. 

Since  there  are  12cZ  in  \s,  in  ISOofZ  there 
are  as  many  shillings  as  12  is  contained  in 
1305<Z  which  is  108^  and  9d  left.  Since 
there  are  20^  in  £1,  in  108*  there  are  as 
many  pounds  as  20  is  contained  in  IO85 
which  is  £5  and  85  left.  Therefore,  130oc^ 
equals  £0  Ss  9d. 


n)lS05d 

20)   108^  9d 

£5  Ss 

Ans.  £5  Ss  9d. 


Note.     For  practice  problems  in  changing  simple  denominate  numbers 
to  higher  denominations  see  par.  24. 


42  RATIONAL  ARITHMETIC 

ILLUSTRATED   SOLUTION 

231.    To  Change  to  Lower  Denomination: 
Problem  :  Reduce  £.575  to  lower  denomination. 

.575£ 
20 


11 


Since  there  are  205  in  £l,  in  £.575  there  are 
ojOyis  20  times  £.575,  which  is   11.5^.     Since  there 

12  are  12c/  in  Is,  in  .5s  there  are  12  times  .5  which 


Q\Od         ^  is  6d. 

Ans.  lis  6d. 

Note.     For  practice  problems  in  changing  denominate  numbers  to  lower 
denominations  see  par.  25. 


ILLUSTRATED   SOLUTION 

232.    To  Change  a   Compound  Denominate   Number 
to  a  Simple  Denominate   Number: 

Problem  :  Reduce  ll6*  6c?  to  a  decimal  of  a  pound. 

\5s 
12)6|0c^  Since  there  are  12c?  in  1^,  in  Qd  there  are  as 

many  shillings  as   12  is  contained  times  in  6 

£|575  which  is  .OS.     \\s  6d  equals  11.5*.     Since  there 

20)  1 1 15005  are  205  in  £l,  in  11.5*  there  are  as  many  pounds 

as  20  is  contained  in  11.55  which  is  £.575. 

Ans.  £.575 

Note.     For  practice  problems  in  changing  compound  denominate  num- 
bers to  simple  denominate  numbers  see  par.  26. 


RATIONxVL  ARITHMETIC  43 

ADDITION   OF   COMPOUND   NUMBERS 
ILLUSTRATED   SOLUTION 

233.    Problem  :  £125  13^  9rf+£23  86'  7d-{-£lS  5d=  ? 

Arrange  the  compound  numbers  so 
that  similar  denominations  fall  in  the 
same  column.  Adding  the  column 
representing  the  lowest  denomination, 
we  have  21c?,  ^Id  equals  Is  and  9^. 
Write  the  9d  under  the  proper  column. 
Carry  Is  to  the  column  of  shillings. 
Adding  we  obtain  22.?  which  equals  £l 

and  25,     Write  25,  carry  £l  to  the  pound  column.     Adding,  we 

have  £162. 


£ 

s 

d 

125 

13 

9 

23 

8 

7 

13 

0 

5 

162 

2 

9 

Ans.   £162  2^  9d. 

SUBTRACTION    OF   COMPOUND   NUMBERS 
ILLUSTRATED   SOLUTION 
234.    Problem  :   Subtract  £15  125  Sd  from  £48  7^  6d, 

Beginning  with  the  column  represent- 
ing the  lowest  denomination,  we  sub- 
tract Sd  from  6d,  which  cannot  be 
performed.  Take  Is  from  7s.  One 
shilling  equals  12c?,  making  18(/  in  all. 
18  minus  8  leaves  lOd.  12^  from  65  can- 
not be  performed.  Take  £l  from  £48. 
£1  equals  205,  plus  6  equals  265,  26.s  minus  12^  equals  145,  £47 
minus  £15  equals  £32,     The  remainder  then  is  £32  145  lOd. 

Note.     For  practice  problems  in  subtraction  of  compound  numbers  see 
pars.  52,  53. 


7 

6 

£4^ 

P 

6^ 

15 

12 

8 

^32 

14 

10 

Ans.   £32  14^  10^. 

44  RATIONAL  ARITHMETIC 

MULTIPLICATION   OF   COMPOUND   NUMBERS 
ILLUSTRATED   SOLUTION 

235.    Problem  :   Multiply  £26  16^  9d  by  23. 

£26         IQs  9d  Beginning  with  the  lowest  denomina- 

23  tion,  multiply  9d  by  23.     This  is  207c?. 

£xQQ      S689      ^01  d  ^^    times    I65    equals    3685.     23    times 

£26  equals  £598. 
£617  5s  Sd  Reduce  207(/  to  shillings   (par.  230) 

Ans    £617  55  M.        ™^l^"^g   ^'^^  ^^-     ^^rite  M  as  part  of 

the  final  product. 

175+3685  =  3855.  Reduce  3855  to  pounds  (par.  230).  Result. 
19c?  55.     Write  5s  as  part  of  the  final  product. 

Add  £19  to  £598.  Result,  £617.  Write  £617  as  the  last  of 
the  final  product. 

DIVISION   OF   COMPOUND   NUMBERS 
ILLUSTRATED   SOLUTION 


236.    Problem  : 

Divide  i 

£26 

I65 

23)617 

23)3855 

46 

23 

157 

155 

138 

138 

£19 

lis 

20 

12 

380^ 

204^ 

5 

3 

9^ 


23)207 
207 

Ans,  £26  I65  9d. 


3855  207^ 

Commencing  with  the  highest  denomination,  divide  £617  by 
23.  This  gives  £26  with  an  undivided  remainder  of  £19.  Write 
the  £26  as  part  of  the  final  quotient. 


RATIONAL  ARITHMETIC  45 

Reduce  £19  to  shillings  by  multiplying  by  20  (par.  229).  This 
equals  3805.  3805  plus  5s  equals  3855.  Divide  3855  by  23,  which 
gives  165,  with  an  undivided  remainder  of  175.  Write  the  I65  as 
part  of  the  final  quotient. 

Reduce  the  175  to  pence  (par.  229). 

This  equals  204c?,  plus  3d  is  207(Z.  207d  divided  by  23  equals 
9c?.  Write  9cZ  as  part  of  the  final  quotient,  making  the  complete 
quotient  £26  I65  9d. 


COMPUTING   TIME 

237.  In  business  it  is  often  necessary  to  compute  the 
interval  of  time  between  two  given  dates. 

Two  methods  are  followed :  Compound  Subtraction 
and  Exact  Days. 


238.    To  Find  the  Time  by  Compound  Subtraction: 

ILLUSTRATED   SOLUTIONS 

Problem  :    Find  the  time  between  October  15,  1908, 
and  July  1,  1916. 

1916        7        1  Use    the    later    date    as    the 

1908      10      15  minuend  and  the  earlier  date  as 

rv        Q      T~^  the  subtrahend.     The  minuend 

will    be    the    1916th    year,    7th 
Ans.   7  yr.  8  mo.  16  da.       month,  and  1st  day.     The  sub- 
trahend will  be  the  1908th  year, 
10th  month,  and  15th  day.     Write  the  subtrahend  beneath  the 
minuend  and  proceed  as  for  compound  subtraction  (par.  234). 

Note.     For  practice  problems  in  finding  the  time  by  compound  sub- 
traction see  par.  52. 


46  RATIONAL  ARITHMETIC 

239.  To  Find  the  Time  in  Exact  Days: 

Problem  :  How  many  days  between  January  8,  1916, 
and  May  19,  1916? 

Jan.      23 

17  K       ciCk     J         V  Subtract  the  date   (8)   from  the 

JjeD.      y4/\)     Lieap  Year  p  n  i  <•    i         •      ,^       n    . 

lull    number    oi    days    m    the    nrst 

Mar.     31  month  (31  days),  leaving  23  days  in 

Apr.      30  January.     February    has     29     days 

May      19  (Leap   Year),   March   31,   April   30, 

132  days  ^^^    May    19.     Write    these    in    a 

column  and  add. 

Ans.  132. 

Note.  For  practice  problems  in  finding  the  time  in  exact  days  see  par. 
53. 

ALIQUOT  PARTS 

240.  The  term  Aliquot  Part  is  applied  to  any  number 
that  is  contained  an  even  number  of  times  in  a  given 
value  or  quantity. 

The  aliquot  parts  of  100  may  be  used  to  great 
advantage  in  many  of  the  arithmetical  operations 
involved  in  business  transactions.  Therefore,  the 
term  aliquot  parts  usually  means  the  aliquot  parts 
of  100. 

241.  The  following  table  shows  the  aliquot  parts  of 
one  hundred  that  are  of  practical  use.  They  are  ex- 
pressed as  common  fractions,  as  decimals,  and  as  cents, 
and  should  be  memorized  and  used  whenever  prac- 
ticable. 


RATIONAL  ARITHMETIC 


47 


Fraction 


X 
2 

3 
3 

4 
4 

1 
5 
2. 
5 
^ 
5 
4. 
5 

i 
6 
5. 
6 

JL 

7 

7 

7 

7 
5. 

7 
6. 

7 

8 
3. 

8 
5. 

8 
7. 
8 


9 
2. 
9 


Decimal 


.0 


.O3 

.U3 


.875 


.Hi 


Cents 


50 


33^ 
66f 


25 

25 

75 

75 

2 

20 

4 

40 

6 

60 

8 

80 

16f 

16f 

m 

m 

14f 

14f 

28-f 

284- 

42f 

42f 

574^ 

571 

71|- 

71f 

85f 

85f 

125 

12i 

375 

m 

625 

621 

87J 


Hi 


Fraction 


A 
9 

5. 
9 

7_ 

9 

8. 

9 

1 
10 

3 
10 

7 
10 

9 
10 


1  1 

2 
1  1 

3 
1  1 

4 
1  1 

5 
1  1 

6 
1  1 

7 
1  1 

8 
1  1 

9 

1  1 
10 
1  1 


1  2 

5 
12 

7 

1  2 
1  1 
12 


16 

3 
16 

5 
1  6 

7 
16 


Decimal 


.441 

.509^ 

.77i 

.889^ 

.1 

.3 

.7 

.9 


•09iV 

.18^ 
.27^ 

•36A 

•54A 

•63t^ 

•72^ 

.81A 


.0^ 

.41f 

.58^ 
.91f 

.06i 
.18f 
.31^ 
.43f 


Cents 

44A 

m 

88f 

10 

30 

70 

90 

9iV 

18T?r 

27T3r 

36A 

45A 
54A 
63Jr 

72tL 
81A 

41f 
58^ 
91t 

6i 

18f 

311 
43f 


48 


Rx\TIONAL  ARITHMETIC 


Fraction 


9 
16 

1  1 
16 

1  3 
16 

1  5 

16 


Decimal 


.56i 
.68f 
.81^ 


Cents 


5Cii 
68f 
81i 

93f 


Fraction 


1 

20 

3 

20 

7 
20 

9 

20 

a 

13 

20 

17 

20 

19 

20 


Decimal 


.05 
.15 
.35 
.45 
.55 
.65 
.85 
.95 


Cents 


5 
15 
35 
45 
55 
65 
85 
95 


ILLUSTRATED   SOLUTION 

Application  of  Aliquot  Parts 

Problem:    Find  the  cost  of  640  lb.  of  tea  at  37^- 

S7U  =  ioi$l 
I  of  640  =  240 

640  lb.  at  $1  a  pound  would  cost  $640.  At  f  of  a  dollar  a 
pound,  640  lb.  would  cost  f  of  $640,  which  is  $240. 

To  avoid  fractions,  first  multiply  by  the  numerator  and  then 
divide  bv  the  denominator. 


Note.     For  practice  problems  in  aliquot  parts  see  par.  27  to  31  inclusive. 


PERCENTAGE 

242.  The  words  per  cent  and  the  name  percentage 
are  derived  from  two  Latin  words,  per  centum,  meaning 
hy  the  hundred. 

243.  Percentage  is  a  system  of  measurement  in 
which  100  is  used  as  the  standard  of  comparison. 

Every  unit  contains  100  one-hundredths  or  100  per 
cent.  The  words  per  cent  are  used  instead  of  the 
denominator,  one-hundredths.  Thus  :  Instead  of  say- 
ing five  one-hundredths,  we  say  5  per  cent.  Nine  per 
cent  would  be  xo^  oi'  -09.  It  could  be  used  in  either 
its  fractional  or  decimal  form. 

244.  The  per  cent  sign  (%)  is  generally  used  instead 
of  the  words  "  per  cent  "  and  instead  of  the  decimal 
point,  just  as  the  cent  sign  (^)  is  used  instead  of  the 
word  '*  cents  "  and  instead  of  the  decimal  point  in  ex- 
pressing United  States  money,  which  is  really  another 
use  of  the  percentage  system. 

(a)  Every  per  cent  may  be  expressed  in  four  different  ways ;  as 
a  common  fraction  with  100  for  a  denominator ;  as  an  equivalent 
common  fraction  in  its  lowest  terms ;  as  a  decimal ;  and  by  the  use 
of  the  sign. 

(6)  The  sign  and  the  decimal  should  never  be  used  together  except 
to  designate  a  part  of  one  per  cent. 

49 


50  RATIONAL  ARITHMETIC 


(c)  25    /o  — .25    —  xbo 


16f%  =  .16 


4 

2_  _      3      _  J, 
3        100        6 

1 


qql(7/  _    qql—       3      _  l 
•^^^  3  /C  —  -^^  3  ~  1  0  0  ~  3 

OQ     07  —    OQ       —23 

-^^     /0~'"'*^      ~  100 

8 

•8    %  =  To  %  =  -008  =  YJ5"  =10  0  0 

245.  In  solving  problems  in  percentage  that  form 
should  he  used  ivhich  makes  the  solution  easiest. 

246.  Several  special  terms  are  used  in  the  subject 
of  percentage.  These  are  Base,  Rate,  Percentage, 
Amount,  and  Difference. 

247.  The  Base  is  the  value,  or  quantity,  represented 
by  100%.     It  is  the  basis  of  comparison. 

248.  The  Rate  is  the  number  of  one-hundredths  used 
in  the  comparison. 

249.  The  Percentage  is  the  value  of  the  rate.  It  is 
the  part  of  the  base  equal  to  the  number  of  hundredths 
represented  by  the  rate.  It  is  the  product  of  the  base 
by  the  rate. 

250.  The  Amount  is  the  base  plus  the  percentage. 

251.  The  Difference  is  the  base  minus  the  percentage. 

252.  In  the  subject  of  aliquot  parts  (pars.  240,  241) 
a  table  showing  the  aliquot  parts  of  one  hundred  is 
given.     The  aliquot  parts  of  100%  are  the  same. 

253.  In  the  following  table  the  fractional  values  of 
the  easier  rates  are  given.  It  will  be  well  to  use  the 
common  fraction  form  for  any  of  these  rates. 


RATIONAL   ARITHMETIC 


51 


-^2  /O  —  4  0 

10  %  =  tV 

334%  =  i 

62i% 

5  %  —  20 

121-%  =  i 

374  %  =  1 

661% 

6i%  =  iV 

161%  =  i 

40  %  =  i 

75  % 

6|%  =  tV 

20  %  =  i 

50  %  =  ^ 

80  % 

83%—  12 

25  %  =  i 

60'%  =  f 

87i% 

5^ 

8 

2^ 
3 

3. 
4 

:4 
5 

7^ 

8 


254.  All  operations  in  percentage  are  solved  in 
accordance  with  the  general  principles  of  multiplica- 
tion and  division.     They  may  be  expressed  as  follows : 

(a)  Base  X  Rate  =  Percentage. 
Percentage  -^  Rate  =  Base. 
Percentage  4- Base  =  Rate. 

(6)  Because  the  smallest  coin  used  in  the  United  States  is  one 
cent,  final  answers  should  be  expressed  in  the  nearest  cent ;  that  is, 
5  mills  (.005)  or  more  would  be  called  another  cent,  less  than  5  mills 
would  be  disregarded. 

255.  To  Find  the  Percentage: 

In  solving  problems  in  this  case  always  use  the  easiest 
method. 

ILLUSTRATED   SOLUTIONS 

Problem  :  Find  37i%  of  $342. 


$342 


3.42 

37i 

171 

23  94 

102  6 

$128.25 

Ans.   $128.25. 

$342  is  the  standard  quantity  or  100%. 
One  per  cent  would  be  -j^q  of  this  which  is 
SA'l  and  37^%  would  then  be  found  by 
multiplying  3.42  by  37^  (par.  135).  The 
result  is  $128.25. 


52  RATIONAL  ARITHMETIC 

Or, 

$342 

cyrvx  Thirty-seven    and  one-half  per  cent  of 


171 


$342  is   the  same  as   .37i  of  342,   which 
means  .37^X342.     Performed  as  explained 
^^  ^^  in  par.  135,  the  result  is  $128.25. 

102  6 


$128.25      Ans.  $128.25. 
Or, 


'*J  I   9.   ft 


$342 

3 

8)1026 


Thirty-seven     and     one-half     per    cent 
equals  |.     |  of  $342  (par.  214)  is  $128.25. 


128.25      Ans.  $128.25. 

All  of  these  methods  will  be  found  to  apply  to  any  problem  in 
which  it  is  necessary  to  find  the  percentage.  In  some  cases  one 
method  will  be  easier  than  another.     Always  use  the  easiest  method. 

256.  Problem  :  A  man  who  is  worth  $8465  has  65% 
of  his  property  in  real  estate,  15%  of  it  in  a  mortgage, 
and  the  remainder  in  cash.     How  much  cash  has  he.^^ 

100%  =  $8465  For   purpose   of   comparison   in   the 

Qo%  4- 15%  =  S0%         above  problem  it  will  be  seen  that  the 

100^  —809"  =209^  man's  entire  property  is  100%.     Sixty- 

oncy  —  1  ^^^  P^^  ^^^^  ^^^  ^^%  (80%)  are  known 

^^      ^  to  be  invested  in  real  estate  and  mort- 

i  of  $8465  =  $1693  gage.     The  remainder  of  the  100%,  or 

20%,  is  in  cash.     Our  problem  then  is 
Ans,   $1693.  to  find  20%  of  the  property,  $8465,  by 

any  of  the  above  explained  methods. 
Since  20%  is  one-fifth,  the  easiest  way  is  to  find  one-fifth  of  $8465, 
which  is  $1693. 


RATIONAL  ARITHMETIC 


53 


257.  Problem:  I  invested  $7460  in  business. 
During  the  first  year  I  gained  25%.  How  much 
did  I  have  invested  in  the  business  at  the  end  of 
that  time? 


4) 


7460 
1865 


9325 
Ans.  $9325. 


Twenty-five  per  cent  is  one-fourth.  One- 
fourth  of  $7469  is  $1865.  If  I  invested  $7460 
and  gained  $1865,  I  must  have  now  $9325. 


Or, 

$7460 
1.25 
37300 
14920 
7460 
$9325.00 

Ans.   $9325. 


The  original  investment  was  100%  of  itself. 
If  it  is  increased  25%,  then  the  new  capital 
would  be  125%  (1.25)  of  the  original  invest- 
ment. Find  125%  of  $7460  by  multiplying  by 
1.25.     Result,  $9325. 


It  is  sometimes  easier  to  combine  the  various  methods  illustrated 
in  the  above  solutions. 


258.    Problem  :   Find  27%  of  $6240. 


4)   6240 


1560 
124.80 


$1684.80 

Ans.  $1684.80. 


Twenty-seven  per  cent  is  made  up  of 
two  easy  rates,   25%  and  2%.     Twentv- 


0- 


five  per  cent  is  \  (1560).  Two  per  cent  is 
twice  1%.  One  per  cent  is  $62.40  and  2% 
is  $124.80  ;  adding  this  amount  to  $1560, 
we  have  $1684.80. 


Note.     For  practice  problems  in  finding  the  pei'centage  see  par.  33. 


54  RATIONAL  ARITHMETIC 

259.    To  Find  the  Base: 

In  solving  problems,  always  use  the  easiest  method. 

ILLUSTRATED   SOLUTIONS 

Problem  :  $1172.34  is  27%  of  what? 

$43142 


27)   1172|34 
108 


92  Since  $1172.34  is  27%,  1%  would  be  Jy  o^ 

g-^  )         $1172.34.     Divide   $1172.34   by   27,  which  is 

^-ya  $43.42.     This  would  be   1%   of  the  original 

value.     One  hundred  per  cent,  or  the  whole 

1——  of  the  original  value,  would  be  100  times  1%, 

^4  which  can  be  found  by  moving  the  figures  in 

54  $43.42    two   places    to    the    left.     Therefore, 

100%  would  be  $4342. 


$43.42  =  1%) 
$4342  =  100% 

Ans,  $4342. 

Or, 

$4342 I 


.27)   1172.34| 

■*^""  $1172.34  is  .27  of  some  number  and  was 

92  found    by   multiplying   the   original   number 

81  by    .27.     In    other    words,    $1172.34    is    the 

]^|3  product  of  one  factor  by  .27.     If  we  divide 

-jQo  $1172.34    (the  product)   by  .27  (one  factor), 

— —7  the  result  will  be  the  other  factor.     Perform 

^^  this  division.     The  result  is  $4342. 
54 

Ans.  $4342. 


RATIONAL  ARITHMETIC  55 

260.  Problem  :  $94.65  is  62i  %  of  what? 

5)    94  65  =  f  Sixty-two   and  one-half  per  cent  is  f  of 

1  qIqq  _  j^  the  required  quantity.     Since  94.65  is  f ,  we 

will  get  ^  of  the  quantity  by  dividing  94.65 

- by  5,  which   is    18.93.     If    18.93   is   ^,    the 

tpl51.44  =  8^  whole  quantity,  or  f,  would  be  8  times  18.93, 

which  is   151.44,  the   value  of  the  original 
Ans.   $151.44.       quantity  or  100%. 

The  problem  given  above  could  be  solved  according  to  either  of 
the  preceding  methods. 

261.  Problem:   $235.64  is  8%  more  than  what? 

$218|185 

1.08)235. 64|000 

216 

196 

108 
ooj^  The  original  quantity  was  100%.    $235.64 

equals     this     and     8%     more.     Therefore, 

$235.64  is  108%  of  what?     Solving  this  by 

any    of   the   above    methods,    we   find    the 


864 


200 

108  original  number  to  be  $218,185,  which,  ex- 

920  pressed   in   the  nearest  equivalent  cent,  is 

864  $218.19. 

"560 

540 

20 

Ans.  $218.19. 

Note.     For  practice  problems  in  finding  the  base  see  par.  34. 


56  RATIONAL  ARITHMETIC 

262.    To  Find  the  Rate: 

In  solving  problems,  always  use  the  easiest  method. 

ILLUSTRATED   SOLUTIONS 
Problem  :   $128.25  is  what  per  cent  of  $342? 

In  the  above  problem  we  simply  want  to 
^|5  =  I  know  what  part  $128.25  is  of  $342. 

'^   3^  371.  $128.2o   is  iffff    of    $342.     Reduced    to 

lowest  terms  |ffff  equals  f .     Then  $128.25 
Ans,   37i%.        is   f    of    $342.    f   equals   37i%.     Therefore, 
>  $128.25  is  37i%  of  $342. 

Or,  Inasmuch  as  $128.25  is  a  certain 

or.2.  per  cent  of  $342,  it  ($128.25)  is  the 

QzLQMQQ  QK  product  obtained  by  multiplying  one 

lOQ  r  ^^^*^^  *^^^^^  desired  rate)  by  another 

I5?_^  factor  ($342).     Therefore,  if  we  di- 

25  65  vide  $128.25    (the  product)   by  342 

23  94  (one  factor) ,  we  shall  obtain  the  other 

171       1  factor     (par.    123).      Perform     this 

349  ~  Q  operation  as  described  in  par.   140. 

The    result    is    .37^,  which    equals 


Arts.  .37ior37i%. 


263.    Problem  :   $9072  is  what  per  cent  of  $7560  ? 

1|20  $9072  is  the  product  obtained  by 

7560)9072|00  multiplying    one    factor    (7560)    by 

another  factor  (the  desired  rate). 
If  we  divide  9072  (the  product)  by 
7560  (one  factor),  we  shall  obtain  the 
other  factor  (par.  123).     The  result, 


7560 

1512  0 
1512  0 


Arts.   1.20  or  120%.       1.20,  equals  120%. 

Note.     For  practice  problems  in  finding  the  rate  see  par.  35. 
For  general  problems  in  percentage  see  par.  36. 


PROFITS  AND  LOSSES 

264.  In  measuring  and  comparing  profits  and  losses, 
it  has  been  found  best  to  do  so  by  means  of  percentage. 

265.  For  this  purpose  it  is  necessary  to  understand  the 
exact  meaning  of  the  special  terms  used  in  connection 
with  this  subject. 

266.  Cost  is  the  value  of  the  investment. 

267.  Prime  Cost  of  an  article  is  the  amount  actually 
paid  for  it.  The  prime  cost  is  sometimes  called  the 
net  cost  and  also  the^r^^  cost. 

268.  The  Gross  Cost  of  an  article  is  the  total  amount 
invested  in  it,  and  includes  the  prime  cost  and  the 
incidental  expenses,  such  as  freight,  cartage,  insurance, 
etc.  The  simple  term  "  cost "  usually  means  the 
gross  cost. 

269.  The  Gross  Selling  Price  is  the  total  amount 
received  for  the  goods  sold. 

270.  The  Net  Selling  Price  is  the  amount  of  the 
gross  selling  price  left  after  incidental  expenses  of  the 
sale  have  been  deducted,  such  as  freight,  commission, 
insurance,  etc. 

57 


58  RATIONAL  ARITHMETIC 

271.  Profit  is  the  difference  between  the  net  seUing 
price  and  the  gross  cost,  ivhen  the  selling  price  exceeds 
the  cost. 

272.  Loss  is  the  difference  between  the  net  selHng 
price  and  the  gross  cost,  when  the  cost  exceeds  the  selling 
price. 

273.  In  the  subject  of  profit  and  loss  : 

Base  =  Gross  Cost 
Rate  =  Rate 
Percentage  =  Profit  or  Loss 
Cost  =100% 

In  solving  problems  in  this  subject,  always  use  the  easiest  method. 

274.  To  Find  Profit,  Loss,  or  Selling  Price: 

ILLUSTRATED   SOLUTIONS 

Problem  :  A  farm  costing  $4200  increased  in  value 
^i%  when  it  was  sold.  Find  the  profit  and  the  selling 
price. 

$4200 

Qgj,  The   cost   is    the    standard    of 

value,     100%.      Then    $4200     is 

100%.       Eight     and      one-third 

per   cent   may   be   found,   as   ex- 

$350.00      Profit  plained  in  par.  255,  to  be  $350. 

4200.00  If  the  cost  is  $4200  and  the  profit 

$4550.00      Selling  Price  is  $350,  the  selhng  price  must  be 

the  sum  of  the  two,  $4550. 


1400 
33600 


Ans. 


\  $350. 
$4550. 


RATIONAL  ARITHMETIC 


59 


Or, 

12)4200 
350 


4550 
Arts. 


Profit 
Selling  Price 
f  $350. 

[$4550. 


Since  8^%  is  -^^,  the  profit  is  yV  of 
the  cost.  One-twelfth  of  $4200  is 
$350.  Added  to  the  cost,  the 
amount  is  $4550. 


275.  Problem:  1200  bushels  of  wheat  were  pur- 
chased at  75  cents  per  bushel  and  later  sold  at  a  loss 
of  17%.  What  was  lost,  and  for  what  was  the  wheat 
sold? 


1200  bu.  @  $.75  =  $900 

$900 
.17 


$153.00     Loss 

$900 
153 

$747     Selling  Price 
$153. 


1200  bushels  of  wheat  at  $.75  cost 
$900  (found  by  the  use  of  aliquot 
parts,  par.  241).  17%  of  $900 
(par.  255)  equals  $153. 

Since  the  goods  cost  $900  and  are 
sold  for  $153  less  than  cost,  they  will 
be  sold  for  $900  minus  $153,  which 
is  $747. 


Ans.    \ 


[  $747. 


Note.     For  practice  problems  in  finding  the  profit  or  loss  and  selling 
price  see  par.  37. 

276.    To  Find  Cost: 

ILLUSTRATED   SOLUTIONS 

Problem  :  My  profit  on  a  certain  transaction, 
figured  at  12^%,  would  be  $720.  What  was  the  cost 
of  the  transaction  ? 


60 


RATIONAL  ARITHMETIC 


5  760 


.125)720.000 
6^5 


95  0 

87  5 

7  50 

7  50 

In  this  problem,  the  rate  is 
12|%  and  the  percentage  $720, 
The  base  equals  the  cost,  which 
may  be  found  as  explained  in 
par.  259  or  260. 


Ans.   $5760. 


0  — i 


Or 

12i 

$720 
8 

$5760  Cost 
Ans.   $5760. 


Twelve  and  one-half  per  cent  equals  ^. 
Since  ^  is  gained  and  the  gain  is  $720,  then 
$720  is  i  of  the  cost.  Therefore,  the  cost 
would  be  8  times  $720. 


277.  Problem  :  I  bought  goods  and  afterwards  sold 
them  at  a  loss  of  23%,  receiving  $412.80  for  them. 
What  did  they  cost? 

100% 

23% 

77% 


5  361103 


.77)412.80  000 
385 


27  8 

23  1 

4  70 

4  62 

80 

77 

300 

231 

The  base,  or  cost,  is  100%,  from  which 
Vq  is  lost,  leaving  77%,   the  measure  of 

the   value  for  which  the   goods   were  sold. 

Then  $412.80  equals  77%  of  the  cost,  which 

may   be   found   as   explained   in   pars.    259, 

260,  261. 


69       Ans.    $536.10. 


RATIONAL  ARITHMETIC 


61 


278.    Problem  :     Goods    are    sold    for    $126,    which 
shows  a  loss  of  lli%.     What  did  they  cost? 


100 
IH 

88f 
126  =  .88f 

$141|75 

8.00)1134|OO 

Or, 

8)126.00 

15.75 
9 


Arts.  $141.75. 


One  hundred  per  cent  is  the 
standard  of  measure,  or  cost ; 
tl^%  has  been  lost.  The 
goods  then  sold  for  88 1%  of 
the  cost.  The  cost  may  be 
found  as  explained  in  par.  261. 


Eleven  and  one-ninth  per  cent  is  equal  to 

If  ^  is  lost,  the  goods 
Therefore,  $126  is  f  of  the  cost. 
If  $126  is  f  of  the  cost,  ^  of  the  cost  would  be 
found  by  dividing  $126  by  8,  which  is  $15.57. 
If  $15.75  is  ^  of  the  cost,  the  whole  cost 


the  common  fraction  ^ 
are  sold  for  |^. 


141.75 

Ans.   $141.75.      would  be  9  times  $15.75,  which  is  $141.75. 
Note.     For  practice  problems  in  finding  the  cost  see  par.  38. 


279.    To  Find  Rate  of  Profit  or  Loss: 

ILLUSTRATED   SOLUTIONS 

Problem  :    Goods  costing  $723.45  are  sold  so  as  to 
gain  $241.15.     What  is  the  gain  per  cent? 
1 


J7^37*5- 
3 


=  -  =  33^ 


3/0 


M^tr^    1 
3 

Ans.  33i%. 


The  cost,  or  base,  is  $723.45.  The 
gain,  or  percentage,  is  $241.15.  Find 
the  rate  as  explained  in  par.  262. 


Or, 


I33i 


723.45)241. 15|00 
217  03  5 

24  11  50 
21  70  35 


2  41  15  Ans,   33i%. 


Use  the  fractional  method 
when  its  reduction  may  be 
determined  at  a  glance ;  use 
the  decimal  method  when 
this  is  not  the  case. 


m  RATIONAL  ARITHMETIC 

280.    Problem  :    Goods  that  cost  $414  are  sold  for 
$492.66.     What  per  cent  is  gained  ? 

$492.66 

414. 

$  78.66 

If  the  goods  cost  $414  and  sell  for  $492.66, 

1^"  the  difference,  $78.66,  must  be  gain  or  profit. 

414)78|66  The  problem  then  is:   $78.66  is  what  per  cent 

41  4  of  $414,  the  cost.'*     This  can  be  ascertained  as 

37  26  explained  in  par.  262. 

37  26 
Ans.  19%. 

Note.     For  practice  problems  in  finding  the  per  cent  of  gain  or  loss  see 
par.  39. 

For  general  problems  in  profit  and  loss  see  par.  40. 


DISCOUNT 

281.  A  Discount  is  an  amount  deducted  from  a  sum 
owed  by  one  person  to  another.  In  measuring  dis- 
counts the  principles  of  percentage  are  used.  There 
are  two  kinds  of  discount,  trade  discount  and  time 
discount. 

282.  Trade  Discount  is  the  discount  allowed  by  a 
manufacturer  or  jobber  to  a  retail  dealer. 

283.  Time  Discount  is  a  discount  allowed  as  a  con- 
sideration for  paying  an  amount  during  a  certain  time. 

TRADE  DISCOUNT 

284.  Manufacturers,  wholesalers,  and  others  doing 
business  of  a  similar  nature,  and  handling  goods  the 
value  of  which  is  likely  to  fluctuate  from  time  to  time, 
have  a  fixed  list  price  for  their  goods.  These  list  prices 
remain  fixed,  and  fluctuations  in  market  rates  are  met 
by  allowing  different  discounts  from  time  to  time ; 
that  is,  if  the  value  drops,  the  discount  is  made  larger, 
and  if  the  price  of  the  goods  rises,  the  discount  is  made 
smaller. 

285.  The  List  Price  of  goods  is  the  price  at  which 
they  are  listed  in  the  catalogue  and  from  which  dis- 
counts are  allowed. 

Goods  are  always  billed  at  the  list  price  and  then  the  discount 
is  deducted  from  the  total. 

63 


64  RATIONAL  ARITHMETIC 

286.  The  Gross  Amount  of  the  bill  is  the  total 
amount  before  any  discounts  have  been  deducted. 

287.  The  Net  Amount  is  the  amount  to  be  paid 
after  all  discounts  have  been  deducted. 

288.  The  Discount  is  the  sum  deducted  from  the 
gross  amount. 

289.  A  Discount  Series  is  several  discounts  deducted 
one  after  another ;   as,  25%,  10%,  and  5%. 

The  first  discount  is  deducted  from  the  gross  amount ;  the  second 
from  the  remainder,  and  so  on ;  the  final  remainder  being  tlie  net 
amount. 

290.  The  principles  of  percentage  are  used  in  per- 
forming all  operations  in  discount. 

Base  =  Gross  Amount 
Rate  =  Rate 
Percentage  =  Discount 
Difference  =  Net  Amount 
Gross  Amount  =  100% 

291.  To  Find  the  Discount  or   Net  Amount: 

Always  use  the  easiest  possible  method. 

ILLUSTRATED   SOLUTIONS 

Problem  :  Goods  listed  at  $420.34  are  sold  at  a 
discount  of  28%.  What  is  the  discount  and  what  is 
the  net  amount  of  the  bill  ? 


RATIONAL  ARITHMETIC  65 


$420.34 

.28 
3362  72 
8406  8 
$117.69  52     Discount 

$420.34 
117.70 
$302.64     Net  Amount 

[$117.70. 
^''^-    ($302.64.    . 

The  gross  amount  of  the  bill, 
$420.34,  is  100%.  Find  28%  of  this 
bv  the  easiest  method.  This  will 
give  $117.70.  The  difference  will 
equal  the  net  amount,  $302.64. 


292.  Problem  :  A  certain  line  of  goods  is  sold  at  a 
discount  of  25%,  20%,  10%,  and  5%.  What  would  be 
the  net  amount  of  a  bill  of  $1214.43,  purchased  under 
these  terms  ?     What  would  the  discount  amount  to  ? 

$1214.43  rpj^jg   problem   involves   a   dis- 

303.60/.^  25%                       count  series.     The  first  discount  is 

910.82^^  deducted  from  the  gross  amount. 

182.16^  20%                       Twenty-five  per  cent  of  $1214.43 

ifoQ  gg  is    $303.61 ;  deducted    from    the 

79  87  100/                       gross     amount     leaves     $910.82. 

^  J —  Twenty  per  cent  of  this,  found  in 

^^^•^^  the  easiest  way,  is  $182.16;     de- 

32.78  5%  ducted,  leaves  $728.66.  Proceed- 
$623.01      Net  Amount         ing  in  the   same  way,  deducting 

$1214.43     Gross  Amount    ^^^^  ^^^  ^^^^  ^^^'  ^^^  "^^  ^^^^^^ 
nr^c  ^-.      TVT  .    A  .         of  the  bill  is  $623.01.     If  the  net 

-^?M1    Net  Amount       ^^^^^^^  ^^  ^^^  ^.^^  .^  ^^,3^^  ^^^ 

$591.42      Discount  the  gross  amount  is  $1214.43,  the 

[  $623  01.  difference  must  be  the  discount, 

^"*-    i  $591.42.  «^91-*2- 

Note.     For  practice  problems  in  finding  the  net  amount  see  par.  41. 


66  RATIONAL  ARITHMETIC 

293.  To  Find  a  Single  Rate  of  Discount  Equal  to  a 
Discount  Series: 

ILLUSTRATED   SOLUTION 

Problem  :  What  single  discount  is  equal  to  a  dis- 
count series  of  25%,  20%,  10%,  and  5%  ? 

4)100%  One  hundred  per  cent  represents  the  gross 

25  amount  of  the  bill.     The  first  discount  is  25% 

or  \  of  this.  One-fourth  of  100%  equals  25%. 
Deduct  the  first  discount  (25%)  from  100%. 
This  leaves  75%  from  which  to  deduct  the 
second  discount.  The  second  discount  is  20% 
or  one-fifth.  One-fifth  of  75%  equals  15%. 
Deduct  this  from  75%.  This  leaves  60%. 
The  next  discount  in  the  series  is  10%  or  one- 
tenth.  One-tenth  of  60%  equals  6%.  60% 
minus  6%  equals  54%.  The  last  discount 
must  then  be  deducted  from  54%.  The 
last  discount  is  5%  or  one- twentieth.  One- 
48.7%  twentieth  of  54%  equals  2.7%.     54%  minus 

4         dft  70/         ^•'^%  equals  51.3%.     This  is  the  net  amount 
*    ^^*       to  be  paid.     Then  the  discount  is  the  differ- 
ence between  the  gross  amount  (100%)  and  the  net  amount  (51.3%). 
This  is  48.7%. 

Note.  For  practice  problems  in  finding  a  single  rate  of  discount  equal 
to  a  discount  series  see  par.  42. 

294.  To  Find  the  Gross  Amount  or  the  List  Price: 

ILLUSTRATED   SOLUTIONS 

Problem  :  On  a  bill  of  goods  sold  at  a  discount  of 
33 i%  the  discount  equals  $55.40.  What  is  the  gross 
amount  of  the  bill  ? 


5) 

75 

15 

10) 

60 

6 

^0) 

54 

2.7 

51.3 

100 

51.3 

RATIONAL  ARITHMETIC 


67 


33i%of  ?  =  $55.40 

$55.40     33^% 
3 


$166.20     100% 
Ans.  $166.20. 


The  discount,  33^%,  is  $55.40.  If 
$55.40  is  33^%,  100%  found  as  ex- 
plained in  par.  259,  is  $166.20. 


295.  Problem  :  A  check  for  $953.80  was  given  in 
full  payment  for  a  bill  of  goods  bought  at  24%  dis- 
count.    What  would  the  gross  amount  of  the  bill  be  ? 

$12  55 


.76)953.80 
76 


193 

152 

418 

38  0 

3  80 

3  80 

The  gross  amount  of  the  bill  is  100%  and 
the  discount  24%.  The  net  amount  of  the 
bill  must  be  76%.  Therefore,  $952.80  equals 
76%  of  the  gross  amount.  Find  the  gross 
amount  as  explained  in  par.  259. 


Ans.  $1255. 

296.  Problem  :  The  net  amount  of  a  bill  of  goods 
sold  at  a  discount  of  40%,  30%,  and  20%  was  $114.24. 
Find  the  gross  amount  of  the  bill. 


100% 
40 
60 

42- 
8.4 
33.6% 


Goods  sold  at  a  discount  of  40%,  30%,  and 
20%  are  sold  at  a  net  price  which  equals  33,6% 
of  the  original  bill  (par.  293).  If  $114.24 
equals  33.6%,  100%,  the  gross  amount,  may  be 
found,  as  explained  in  par.  259,  to  be  $340. 


68 


RATIONAL  ARITHMETIC 


$340 


.336)114.240 
100  8 
13  44 
13  44 

Ans.  $340. 

Note.  For  practice  problems  in  finding  the  gross  amount  or  the  list 
price  see  par.  43. 

297.    To  Find  the  Rate  of  Discount: 

ILLUSTRATED   SOLUTIONS 

Problem  :  The  gross  amount  of  a  bill  of  goods  is 
$346.40.  The  net  amount  is  $259.80.  What  is  the 
rate  of  discount  ? 

$346.40 
259.80 


$86.60 


The  difference  between  the  gross  amount 

25     and    the    net    amount    equals    the    discount, 

346  40)86.60  00     which    is    $86.60.     The    problem    then     is : 

6Q^8  0        $86.60  is  what  per  cent  of  $346.40.?     This  is 

iiWoo  ^'''''''^ ^"^ ^^ ^^^"^  ^^^''  ^^^^* 

17  32  00 


or 

1 

B600r_l 

^04^     4 
4 

Ans.  25%. 


or  25%. 


Note.     For  practice  problems  in  finding  the  rate  of  discount  see  par.  44. 


RATIONAL  ARITHMETIC  69 

298.  To  Find  What  Price  to  Mark  Goods  in  Order  to 
Allow  a  Certain  Discount  and  Still  Make  a  Certain  Profit: 

ILLUSTRATED   SOLUTION 

Problem  :  What  price  must  we  mark  goods  costing 
$*214  in  order  that  we  may  allow  a  discount  series  of 
20%,  10%,  and  5%  and  still  make  a  profit  of  25%  ? 

$214 
53.50 

$267.50 

100% 
20 

80 
g  By  the  principles  of  profit  and  loss,  we  see 

^  that  to  make  a  profit  of  25%,  goods  costing 

o  ^  $214   must   be   sold   for   $267.50    (par.   274). 

— - — ^  Then  we  must  sell  the  goods  for  a  net  amount 

68.4%  Qf    $267.50.     If    a    discount    series    of    2q^c, 

$391 108     10%,  and  5%  is  to  be  allowed  from  the  gross 

684^267  500 loo     ^^ount  of  the  bill,  the  net  amount  of  the  bill 

oQx  Q  will  be  68.4%  of  the  gross  amount  (par.  293). 

Therefore,  $267.50  equals  68.4%  of  the  gross 

amount,  or  asking  price,  which  will  be  found, 

as  explained  in  par.  259,  to  be  $391.08. 


62  30 
61  56 


740 
684 


56  00 

54  72 


128 

Ans.   $391.08. 

Note.     For  practice  problems  in  finding  what  price  to  mark  goods  in 
order  to  allow  a  certain  discount  and  still  make  a  certain  profit  see  par.  45. 
For  general  problems  in  trade  discount  see  par.  46. 


COMMISSION  AND  BROKERAGE 

299.  A  Commission  Merchant  is  a  person  or  firm  who 
buys  or  sells  merchandise  for  another  person  or  firm. 

A  commission  merchant  actually  handles  the  goods  and  buys 
and  sells  them  as  if  for  himself,  but  in  reality  for  another  person  or 
firm. 

300.  A  Broker  is  a  person  or  firm  who  arranges 
transactions  between  other  persons. 

A  broker  does  not  handle  the  merchandise  himself,  but  simply 
brings  the  buyer  and  seller  together  in  the  interest  of  one  or  the 
other. 

301.  The  Principal  is  the  person  or  firm  for  whom 
the  business  is  transacted. 

302.  The  Commission  is  the  compensation  allowed 
the  commission  merchant  or  the  broker. 

303.  The  Gross  Proceeds  of  a  sale  or  collection  is 
the  entire  amount  received  from  the  purchaser  or 
debtor  by  the  commission  merchant. 

304.  The  Charges  are  the  incidental  expenses  of  the 
sale  or  purchase. 

305.  The  Net  Proceeds  is  the  amount  remaining 
after  the  charges  have  been  deducted  from  the  gross 
proceeds.  It  is  the  amount  to  be  returned  by  the 
commission  merchant  to  his  principal. 

70 


RATIONAL  ARITHMETIC  71 

306.  Prime  Cost  is  the  first  cost  of  goods  purchased 
by  the  commission  merchant  in  the  interest  of  his 
principal. 

307.  The  Gross  Cost  is  the  prime  cost  plus  the 
charges  incidental  to  the  purchase. 

308.  Account  Sales  is  an  itemized  statement  of 
sales  of  merchandise  by  a  commission  merchant.  It 
shows  the  amount  for  which  the  goods  were  sold,  the 
charges,  and  the  net  proceeds  of  the  sale. 

309.  Account  Purchase  is  an  itemized  statement 
covering  the  merchandise  purchased  by  a  commission 
merchant  and  shows  the  prime  cost  plus  the  charges. 

310.  In  solving  problems  in  commission,  the  general 
principles  of  percentage  are  used. 

Base  =  Gross  Sales  or  Prime  Cost 
Rate  =  Rate  of  Commission 
Percentage  =  Commission 
Net  Proceeds  =  Difference 
Gross  Cost  =  Amount 
Gross  Sales  or  Prime  Cost  =  100% 

311.  To  Find  the  Commission  and  Net  Proceeds  or 
Gross  Amount  of  Purchase : 

ILLUSTRATED   SOLUTIONS 

Problem  :  A  commission  merchant  sold  goods  for 
$2346.40.  His  commission  was  2^%.  Charges  for 
insurance,  freight,  etc.,  amounted  to  $98.40.  Find 
the  commission  and  net  proceeds.  • 


72 


RATIONAL  ARITHMETIC 


$2346.40 

58.66 

98.40 

157.06 

$2189.34 

Ans. 


2i%  Com- 
mission 
Charges 

Net  Proceeds 

$58.66. 
$2189.34. 


Gross  sales,  100%,  is  $2346.40. 
Two  and  one-half  per  cent  of  this, 
the  commission,  is  $58.66.  Since 
the  commission  merchant  spent 
$98.40  for  expenses  and  kept  $58.66 
for  commission,  the  sum  of  these, 
$157.06,  must  be  deducted  from 
$2346.40,  leaving  a  net  proceeds 
of  $2189.34. 


312.  Problem  :  A  commission  merchant  bought 
500  barrels  of  apples  at  $2.75  on  a  commission  of  5% 
and  paid  $15  for  cartage  and  $52.50  for  cooperage. 
For  what  sum  must  his  principal  write  a  check  to  cover 
the  entire  transaction  ? 


500  bbl.  @  $2.75  =  $1375 
$1375 


68.75 

15. 

52.50 


5% 


$1511.25 
Ans.  $1511.25. 


Five  hundred  barrels  of  apples 
at  $2.75  cost  $1375.  The  com- 
mission, 5%,  equals  $68.75.  The 
principal  will,  therefore,  have  to 
send  the  commission  merchant 
$1375  to  pay  for  the  goods,  $68.75 
for  his  commission,  $15  to  pay  for 
cartage,  and  $52.50  for  cooperage, 
or  $1511.25  in  all. 


Note.     For  practice  problems  in  finding  the  commission  and  net  pro- 
ceeds or  gross  amount  of  purchase,  see  pars.  47  and  48. 

313.    To  Find  the  Gross  Sales  or  Net  Purchase  Price: 


ILLUSTRATED   SOLUTIONS 


Problem  :  A  commission  merchant  working  on  2^% 
commission  received  $134.50  for  selling  a  consignment 
of  flour.     What  did- the  flour  sell  for.^^ 


RATIONAL  ARITHMETIC  73 


$134.50  =  2i% 
$5  380 


.025)134.500 

125  $134.50  is  ^%.     Find   100% 

as  explained  in  par.  259. 


Ans.  $5380. 


Or, 

QJ-Or  —  JL 
^2/0  —  4  0 

134.50  =  A 
40 


95 

75 

2  00 

2  00 

$5380.00  Ans.  $5380. 

314.  Problem  :  The  net  proceeds  is  $568.40.  The 
charges  for  freight,  insurance,  etc.,  are  $27.40.  The 
commission  is  3%.     For  what  were  the  goods  sold  ? 


$568.40 
27.40 


$595.80 

6  14  22 

.97)595.80  00 

582 

13  8 
97 

4  10 

3  88 

22  0 
19  4 

2  60 
194 

The  net  proceeds  is  $568.40.  Charges  for 
freight,  insurance,  etc.,  are  $27.40;  added  to 
$568.40  equals  $595.80.  This  is  the  amount 
remaining  from  the  sales  after  the  commissioin 
alone  has  been  deducted.  If  the  commission 
is  3%,  then  $595.80  is  97%.  Find  the  total 
amount  of  the  sales  as  explained  in  par.  259, 
which  is  $614.22. 


Ans.  $614.22. 


74 


RATIONAL  ARITHMETIC 


315.  Problem  :  I  sent  a  commission  merchant  $927 
to  invest  in  apples.  How  many  barrels  at  $3.75  can 
be  purchased  after  deducting  a  commission  of  3%? 


9  00 


1.03)927.00 
927 

2  40 


3.75)900.00 
750 
150  0 
150  0 

Ans.  240. 


$927  includes  the  amount  of  purchase  and 
the  commission  of  3%.  Therefore,  $927  is 
103%.  100%  may  be  found,  as  explained  in 
par.  261,  to  be  $900. 

If  one  barrel  of  apples  cost  $3.75,  for  $900  we 
can  buy  as  many  barrels  as  $3.75  is  contained  in 
$900,  which  is  240. 


Or, 


3%  of  $3.75  =  .1125,  commission  on  1  bbl. 

$3.75         Cost   of  1  bbl. 

.1125     Commission  on  1  bbl. 
$3.8625     Gross  cost  of  1  bbl. 


240 


3.8625)927.0000 
772  50 
154  500 
154  500 

Ans.   240. 


Since  the  market  price  of  1  bbl.  is  $3.75, 
the  commission  on  one  barrel  would  be  3% 
of  $3.75  which  is  .1125,  and  the  gross  cost 
of  one  barrel  would  be  $3.8625.  For  $927 
the  commission  merchant  could  buy  as  many 
barrels  as  $3.8625  is  contained  in  $927,  which 
is  240. 


Note.     For  practice  problems  in  finding  the  gross  sales  or  net  purchase 
price  see  par.  49. 


RATIONAL  ARITHMETIC  75 

316.    To  Find  the  Rate  of  Commission: 

ILLUSTRATED   SOLUTION 

Problem  :  A  commission  merchant  charges  $80.16 
for  selling  a  bill  of  goods  for  $1336.  What  is  his  rate 
of  commission  ? 

|06 

lSS6')80ll6  Since  the  commission  is  figured  as  a  certain 

en  1 «  per  cent  of  the  amount  of  sales,   this  problem 

— really   is:    $80.16   is   what  per   cent  of   $1336, 

which,  solved  as  explained  in  par.  262,  is  .06  or  6%. 
Ans.  6%. 

Note.  For  practice  problems  in  finding  the  rate  of  commission  see 
par.  50. 

For  general  problems  in  commission  see  par.  51. 


INTEREST 

317.  Interest  is  the  amount  paid  for  the  use  of 
money. 

(a)  When  the  use  of  real  estate  is  allowed  to  someone  other  than 
the  owner,  the  compensation  is  called  rent ;  the  compensation  for 
the  use  of  personal  property  is  called  hire;  the  compensation  for 
the  use  of  manual  labor  is  called  wages;  the  compensation  for  the 
use  of  mental  labor  and  time  is  called  salary. 

(b)  The  amount  of  interest  to  be  paid  depends  upon  the  time 
that  the  money  is  used,  the  sum  that  is  used,  and  the  way  it  is 
used  (risk  involved). 

(c)  The  first  two  are  self-fixing.  The  third  is  subject  to  agree- 
ment between  the  parties.  If  there  is  danger  of  the  original  sum 
being  lost,  a  larger  rate  should  be  paid  for  its  use.  Also  when 
money  is  plentiful  and  easy  to  hire,  the  rate  should  be  lower  than 
if  it  were  scarce  and  hard  to  hire. 

(d)  It  has  been  found  that  the  best  way  of  figuring  interest  is  on 
a  percentage  basis.  Therefore,  the  principles  of  percentage  with 
another  element,  time,  govern  the  subject  of  interest. 

318.  Principal  is  the  sum  for  the  use  of  which  interest 
is  charged. 

319.  Rate  is  the  per  cent  of  the  principal  charged 
for  the  use  of  the  principal  for  one  year. 

320.  The  Legal  Rate  is  the  rate  fixed  by  law  to  be 
understood  when  no  rate  is  mentioned  by  the  parties. 
It  differs  in  different  states.  The  legal  rate  in  a 
majority  of  the  states  is  6%. 

76 


RATIONAL  ARITHMETIC  77 

Charging  more  than  a  reasonable  compensation  is  called  usury. 
Some  of  the  states  name  a  definite  maximum  rate ;  in  such  states 
to  charge  more  is  usury.  Where  no  maximum  rate  is  fixed  by  law, 
it  is  a  question  for  the  courts  to  decide  whether  or  not  usury  is 
charged  in  a  given  case. 

321.  Time  is  the  period  for  which  the  principal  is  used. 

322.  The  Amount  is  the  sum  of  the  principal  and 
interest. 

323.  The  interest  on  $1  for  one  year  at  6%  would 
be  6%  of  $1,  which  is  $.06.  The  interest  on  $1240 
for  one  year  at  6%  would  be  6%  of  $1240,  which  is 
$74.40.  The  interest  for  one  year,  then,  is  always 
equal  to  the  percentage  of  the  principal  represented  by 
the  rate. 

324.  The  general  principle  on  which  all  interest  is 
based  is  : 

Principal  X  Rate  X  Time  =  Interest 

In  figuring  the  interest  for  part  of  a  year,  two  dif- 
ferent methods  arise  :  Accurate  Method  and  Ordinary 
Method. 

ACCURATE  INTEREST 

325.  Accurate  Interest  gives  a  year,  or  any  part  of 
a  year,  its  exact  value. 

(a)  In  computing  the  accurate  interest  for  any  part  of  a  year  the 
time  is  counted  in  days,  and  each  day  given  its  actual  value,  3^^  of 
a  year. 

(6)  This  method  is  used  only  in  figuring  interest  on  United  States 
bonds,  on  foreign  moneys,  and  by  special  agreement. 

326.  Accurate  Interest  is  found  by  applying  the 
general  principle  of  interest  with  absolute  accuracy. 


78  RATIONAL  ARITHMETIC 

ILLUSTRATED   SOLUTIONS 

327.    Problem  :    Find  the  accurate  interest  of  $1140 
for  93  days  at  6%. 

(h-,  1  j^Q  Six  per  cent  of  $1140  equals  $68.40.     The  interest  on 

$1140  for  one  year  is  $68.40.     The  mterest  for  93  days 

^ —      equals  -^^^  of  $68.40,  which,  found  as  explained  in  par. 

$68.40      214,  is  $17.43. 

Find  ^^5  of  $68.40. 


$68.40 

$17  427 

93 

365)6361  200 

205  20 

365 

6156  0 

2711 

$6361.20 

2555 

156  2 

146  0 

10  20 

7  30 

2  900 

2  555 

345 

Ans.   $17.43. 

328.    Problem  :    Find  the  accurate  interest  of  $1140 

for  458  days. 

458  days  =  1  year  93  days. 

*  This   problem    differs    from   the  preceding 

^  '  •'*'^  one  only  in  the  matter  of  time.     The  interest 

$85.83  for   one    year   is    $68.40.     For    93    days    the 

Ans.   $85.83.      interest,  found  as  in  the  previous  problem,  is 

$17.43.     The  sum  of  the  two  will  equal  the 
interest  for  1  year  and  93  days. 


RATIONAL  ARITHMETIC  79 

ORDINARY   INTEREST 

329.  Ordinary  Interest  Method  is  that  used  by  busi- 
ness men  under  ordinary  circumstances. 

It  differs  from  exact  interest  simply  in  that  one  day  is  roughly 
considered  one  three-hundred-sixtieth  of  a  year. 

This  is  arrived  at  in  this  way  :  one-twelfth  of  a  year  is  called  one 
month,  and  one-thirtieth  of  a  month  is  called  a  day.  So  that  one 
day  is  one-thirtieth  of  one-twelfth,  or  one  three-hundred-sixtieth  of 
a  year. 

330.  To  figure  the  ordinary  interest  for  parts  of  a 
year,  business  men  sometimes  count  the  time  in  months 
and  days,  considering  each  month  to  have  thirty  days ; 
sometimes  the  time  is  figured  in  exact  days. 

The  latter  plan  is  always  followed  by  banks.  Some  authorities 
divide  ordinary  interest  into  two  classes.  Common  and  Bankers' . 

331.  The  best  method  of  calculating  ordinary  in- 
terest is  by  what  is  known  as  the  Sixty-day  Method. 

By  this  method  the  interest  is  always  found  at  6% 
first  and  then  changed  to  the  rate  desired. 

EXPLANATION 

At  6%,  the  interest  on  any  sum  for  one  year  would 
be  six  one-hundredths  of  the  principal.  One  one- 
hundredth  of  the  principal,  then,  would  be  the  interest 
at  6%  for  one-sixth  of  a  year.  One-sixth  of  a  year  is 
2  months,  or  60.  days.  Then  the  interest  on  any 
principal  for  60  days,  or  2  months,  at  6%,  would  be 
one  one-hundredth  of  itself.  Thus  the  interest  at  6% 
on  $2420  for  60  days  is  $24.20.  For  12  days  it  would 
be  one-fifth  of  the  interest  for  60  days,  or  $4.84,  and 


80  RATIONAL  ARITHMETIC 

so  on.  The  interest  at  1%  would  be  one-sixth  of  the 
interest  at  6%,  or  $4.0333  for  60  days.  The  interest 
at  3i%  would  be  3i  times  $4.0333,  which  is  $14.11  for 
60  days. 

SIXTY-DAY   METHOD  —  ORDINARY   INTEREST 

RULE 

332,  Write  the  principal.  Set  off  the  interest  for 
2  months,  or  60  days,  at  6%.  Using  the  interest  for 
60  days  as  a-  basis,  the  interest  at  6%  for  any  other 
period  may  be  easily  ascertained.  When  the  interest 
for  the  desired  period  has  been  found  at  6%,  change  to 
the  rate  desired. 

Interest  in  partial  results  should  always  be  carried  to  the  fourth 
decimal  place. 

ILLUSTRATED   SOLUTIONS 

333.  Problem  :  Find  the  interest  on  $1246.40  for 
1  year  9  months  16  days  at  6% ;    at  4%. 

$1246.40  1  yr.  9  mo.  16  da.  at  6% ;   at  4% 

12.4640  Int.  for  60  da.  at  6% 

74.7840  Int.  for  1  yr.  at  6% 

49.8560  Int.  for  8  mo.  at  6% 

6.2320  Int.  for  1  mo.  at  6% 

2.0773  Int.  for  10  da.  at  6% 

1.2464  Int.  for  6  da.  at  6% 

6)134.1957  Int.  for  1  yr.  9  mo.  16  da.  at  6% 

22.3659  Int.  for  1  yr.  9  mo.  16  da.  at  1% 

4 

$89.4636  Int.  for  1  yr.  9  mo.  16  da.  at  4% 


RATIONAL  ARITHMETIC  81 

Set  off  the  interest  for  2  months  or  60  days  at  6%,  $12.4040. 
We  now  find  the  interest  for  1  year,  8  months  and  1  month,  10  days 
and  6  days.  One  year  is  6  times  2  months.  Eight  months  is  4 
times  2  months.  One  month  is  one-half  of  60  days.  Ten  days  is 
one-sixth  of  60  days.     Six  days  is  one-tenth  of  60  days. 

12.4640X  6  =  $74.7840,  ^Yhich  is  interest  for  1  yr.  at  6% 
12.4640  X  4  =  $49.8o60,  which  is  Interest  for  8  mo.  at  6% 
12.4640^  2  =  $6.2320,  which  is  interest  for  1  mo.  at  6% 
12.4640^  6  =  $2.0773,  which  is  interest  for  10  da.  at  6% 
12.4640  ^10  =  $1.2464,    which  is  interest  for  6  da.  at  6% 

Adding  these  partial  results,  we  have  a  total  of  $134.1957,  which 
is  the  interest  for  1  year  9  months  16  days  at  6%.  Dividing  this 
by  6  gives  us  the  interest  at  1%,  which  is  $22.3659.  Multiplying 
this  by  4  gives  $89.4636,  which  is  the  interest  for  1  year  9  mouths 
16  days  at  4%. 

Or, 

$134  1957  at  69^  Divide  the  interest  at  6%  by  3,   which 

^      gives  $44.7319,  which  is  the  interest  at  2%. 
Subtract  this  from  the  interest  at  6%.     The 


44.7319  at  2% 


$89.4638  at  4%      difference  is  $89.4638,  the  interest  at  4%. 

334.    Problem  :    What  is  the  interest  on  $428.75  for 
214  days  at  6%  ? 


$428.75 

214  da.. at  6% 

4.2875 

Int.  for  60  da.  at  6% 

12.8625 

Int.  for  180  da.  at  6% 

2.1437 

Int.  for  30  da.  at  6% 

.2143 

Int.  for  3  da.  at  6% 

.0714 

Int.  for  1  da.  at  6% 

$15.2919 

Int.  for  214  da.  at  6% 

ns.  $15.29. 

82  RATIONAL  ARITHMETIC 

In  this  problem,  the  most  convenient  periods  of  time  would  be 
180-30-3-1  days.  Set  off  the  interest  for  60  days,  which  is  $4.2875. 
One  hundred  eighty  days  is  3  times  60.  Thirty  days  is  one-half 
of  60.  Three  days  is  one-tenth  of  30.  One  day  is  one-third  of 
3  days.  Adding,  gives  us  $15.2919,  the  interest  for  214  days 
at  6%. 

Note.     For  practice  problems  in  finding  the  interest  see  par.  54. 

335.  Problem  :  Find  the  Interest  on  $1234.28  from 
January  3,  1915,  to  August  1,  1916,  at  6%. 


1916 

8 

1 

$1234.28 

1  yr.  6  mo.  28  da.  at  6% 

1915 

1 

3 

12.3428 

Int.  for  60  da.  at  6% 

1 

6 

28 

74.0568 

37.0284 

4.9368 

.8228 

Int.  for  1  yr.  at  6% 
Int.  for  6  mo.  at  6% 
Int.  for  24  da.  at  6% 
Int.  for  4  da.  at  6% 

$116.8448 

Int.  for  1  yr.  6  mo.  28  da. 

at  6% 

Ans.  $116.8448. 

In  this  problem  it  is  necessary  to  find  the  time  between  Janu- 
ary 3,  1915,  and  August  1,  1916.  This  will  be  found  (par.  238)  to 
be  1  year  6  months  and  28  daj's. 

One  year  is  6  times  2  months,  6  months  is  3  times  2  months, 
24  days  is  4  times  6  days  (6  days  being  one- tenth  of  60  days). 
4  days  is  one-sixth  of  24.  The  total  is  $116.8448,  which  is  the 
interest  at  6%  for  1  year  6  months  and  28  days. 

336.  Problem  :  Find  the  interest  on  $514.95  from 
January  18,  1915,  to  June  7,  1915,  at  4i%.  Time 
computed  in  exact  days. 


RATIONAL  ARITHMETIC  83 


Jan.      13  $514.95  140  da.  at  4>i% 

Int.  for  60  da.  at 


Feb.  28 
Mar.  31 
Apr.  30 

5.1495 
10.2990 
1.7165 

Mav  31 
June   7 

4)12.0155 

3.0038 

140  days 
Ans.   $9.0117. 

$9.0117 

0 

0 
0 


Int.  for  120  da.  at 
Int.  for  20  da.  at 
Int.  for  140  da.  at  6% 
Int.  for  140  da.  at  li% 
Int.  for  140  da.  at  4^% 


Find  the  exact  number  of  days  between  January  18,  1915,  and 
June  7,  1915  (par.  239),  which  is  140  days.  This  consists  of  120 
days  and  20  days.  One  hundred  twenty  days  is  twice  60  days. 
Twentv  davs  is  one-third  of  60  davs,  making  the  total  interest 
$12.0155  for  140  days  at  6%.  The  difference  between  6%  and 
4^%  is  1^%.  One  and  one-half  per  cent  is  one-fourth  of  6%. 
Divide  $12.0155,  the  interest  at  6%,  by  4,  which  gives  us  $3.0038, 
the  interest  at  1^%.  Subtract  this  from  the  interest  at  6%,  which 
leaves  $9.0117,  the  interest  at  4^%. 

Note.  For  practice  problems  in  finding  the  interest  when  the  time  has 
to  be  found  either  in  exact  days  or  by  compound  subtraction  see  par.  55. 

SIXTY-DAY   METHOD  —  ACCURATE   INTEREST 

337.  The  difference  between  ordinary  interest  and 
accurate  interest  for  any  part  of  a  year  is  one  seventy- 
third  of  the  ordinary  interest. 

338.  The  interest  for  one  year  or  any  number  of 
years  is  the  percentage  yalue  of  the  rate  and  is  the  same 
for  ordinary  and  accurate  because  it  is  the  standard  for 
both  methods. 

339.  Ordinary  interest  for  any  part  of  a  year  is 
greater  than  accurate  interest  for  the  same  time.  • 


84 


RATIONAL  ARITHMETIC 


340.  To  change  the  ordinary  interest  for  any  part  of 
a  year  to  the  accurate  interest  for  the  same  time, 
divide  the  ordinary  interest  by  73.  Subtract  this  from 
the  ordinary  interest. 

341.  Problem  :  Find  the  accurate  interest  on  $246 
for  73  days  at  6%. 


$240. 


2.40 

.48 
.04 

73)2.92 
.04 


73  da.  at  yjyo 
Int.  for  60  da.  at 
Int.  for  12  da.  at 
Int.  for  1  da.  at 
Ordinary  interest 


0 


0 


c 


2.88     Accurate  interest 
Ans.  $2.88. 


First  find  the  ordinary 
interest,  as  explained  in 
par.  334,  on  $240  for  73 
days  at  6%,  which  is  $'2.92. 

One  seventy-third  of 
$2.92  is  $.04.  The  ordi- 
nary interest,  therefore, 
is  $.04  larger  than  the  ac- 
curate interest,  which  is 
found  by  subtracting  $.04 
from  $2.92. 


342.    Prohlem  :    Find  the  accurate  interest  for  41^ 
days  at  4i%  on  $425. 

419  days  =  l  year  and  54  days 

$425. 


0 


4>.^5 


2.125 
1.700 

73)3.825 
.0523 
3.7727 
25.50 
4)29.2727 
7.3181 
$21.9546 
Ans.  $21.95. 


0 


0 


1  yr.  54  da.  at  4^' 
Int.  for  60  da.  at 
Int.  for  30  da.  at 
Int.  for  24  da.  at  uyo 
Ordinary  interest  for  54  da. 

Accurate  interest  for  54  da.  at 
Accurate  interest  for  1  yr.  at 


o 


0 


RATIONAL  ARITHMETIC  85 

This  problem  differs  from  tlie  previous  one  in  that  the  time  is 
more  than  a  year.  Three  hundred  sixty-five  days  should  be  sub- 
tracted from  419,  leaving  54  days  more  than  a  3''ear,  so  that  the  full 
time  is  1  year  and  54  days. 

Find  the  interest  by  the  ordinary  interest  method  for  54  days  (par. 
334).  This  is  $3,825  at  6%.  Change  this  to  accurate  interest  as  ex- 
plained above.     The  accurate  interest  for  54  days  at  6%  is  $3.7727. 

Six  per  cent  of  $425  gives  the  interest  for  one  year  (both  accurate 
and  ordinary).  Six  per  cent  of  $425  is  $25.50.  Add  this  to  the 
accurate  interest  for  54  days  and  we  have  the  accurate  interest 
for  1  vear  and  54  davs,  which  is  $29.2727. 

Dividing  this  by  4  gives  the  interest  at  1^%,  which  is  $7.3181. 
Deduct  this  from  the  accurate  interest  at  6%  and  we  have  $21.9546, 
the  accurate  interest  for  1  year  and  54  days  at  4^%. 

343.  Prohlem  :  Find  the  accurate  interest  on  £214 
\\s  9d  for  115  days  at  5%. 


12)9 


11 


20)11 


d 


75s 


75s 


5875£ 

£214.5875     115  da.  at  5% 
2.1458     Int.  for  60  da.  at 


2.1458  Int.  for  60  da.  at 

1.0729  Int.  for  30  da.  at  6% 

.7152  Int.  for  20  da.  at  6% 

.1788  Int.  for  5  da.  at  6% 


73)£4.1127  Ordinary  interest  at  6% 
.0563 

4.0564  Accurate  interest  at  6% 

.6760  Accurate  interest  at  1% 


£3.3804     Accurate  interest  at  5% 


86  RATIONAL  ARITHMETIC 

Reduce  £'^214   lis  9d  to  pounds   (par. 

£3804  .608  ^32),    which    is     £214.5875.     Find    the 

QQ  TQ  ordinary  interest  (par.  334).     Change  to 

^  „„„^      ^  ^^^  7       accurate    interest    by    subtracting    one 

7.60805  7.29Da  +    +i  •  i    ^  •.    u  /        onx     a.,. 

seventy-third  oi  itseli   (par.  341).      Ihis 

Ans.   i/O   tS  la.  equals    £4.0564,    which   is   the   accurate 

interest  at  6%.     Subtract  one-sixth  (par. 
333).     This  equals  accurate  interest  at  5%. 

Note.     For  practice  problems  in  accurate  interest  see  par.  56. 

344.    The  best  combinations  for  finding  the  interest 
by  the  sixty-day  method  for  from  one  to  thirty  days : 

1  day   =  eV  of  60,  or  ^  of  6 

2  days  =  3V  of  60,  or  i  of  6 

3  days  =  2V  of  60,  or  i  of  6 

4  days  =  3  days  +  1  day 

5  days  =  iV  of  60  days 

6  days  =  ro  of  60  days 

7  days  =  6  days  +  1  day 

8  days  =  6  days +  2  days 

9  days  =  6  days +  3  days 

10  days  =  i  of  60  days 

11  days  =  10  days  +  1  day 

12  days  =  i  of  60  days 

13  days  =  10  days +  3  days 

14  days  =  12  days  +  2  days 

15  days  =  i  of  60  days  or  i  of  30 


9 

RATIONAL  ARITHMETIC  87 

16  days  =  10  days  +  6  days 

17  days  =  15  days +  2  days 

18  days  =  12  days  +  6  days  or  3X6 

19  days  =  15  days  +  3  days  +  1  day 

20  days  =  |^  of  60  days 

21  days  =  20  days  +  1  day 

22  days  =  20 +  2 

23  days  =  20+3 

24  days  =  20  +  4  or  4X6 

25  days  =  20 +  5 

26  days  =  20 +  6 

27  days  =  24 +  3  (i  of  24) 

28  days  =  24 +  4  (i  of  24) 

29  davs  =  24+5 

30  days  =  i  of  60 

345.    To  Find  the  Interest  at  Common  Rates  Other  Than 

0' 

\%  =  A  of  6%  2%  =  i  of  6%  5%  =  6%  - 1% 

i%  =  \  of  i%  2i%  =  2% + ¥7o  7%  =  6%  + 1% 

1%  =  4+1%  3%  =  i  of  6%  8%  =  6%+2% 

1%  =  i  of  6%  4%  =  6%  -  2%  9%  =  6% + 3% 

H%  =  1  of  6%  4i%  =  6%  -  U%  10%  =  lOX  1% 


88 


RATIONAL  ARITHMETIC 


346.  To  Find  the  Time  :  Divide  the  given  interest 
by  the  interest  on  the  principal  for  one  year,  at  the 
given  rate. 

(a)  The  divisor  should  be  carried  to  the  fourth  decimal  place  if 
necessary. 

(6)  The  quotient  should  be  carried  to  the  fourth  decimal  place. 

(c)  In  the  final  result,  more  than  half  should  be  considered 
another  day. 

ILLUSTRATED   SOLUTIONS 

347.  Problem  :  In  what  time  wih  $532.56  produce 
$48,  interest  at  5%  ? 

$532.56 
.05 


$26.628C 

1 

1 

8026 

26.628)48.000 

0000 

26  628 

21  372  0 

21  302  4 

69  600 

53  256 

16  3440 

15 

9768 

3672 


1.8026  yr. 

12 

9.6312  mo. 

30 

18.9360  da. 


Five  per  cent  of  $532.56  is 
the  interest  on  $532.56  for  one 
year.     It  is  $26.6280. 

If  $532.56  earns  $26.6280  in 
one  year,  it  will  take  as  manv 
years  to  earn  $48  as  $26.6280 
is  contained  in  $48,  which  is 
1.8026  years. 

Reducing  1.8026  j^ears  to 
years,  months,  and  days  (par. 
231),  we  have  1  year  9  months 
and  19  days. 


Ans.  1  yr.  9  mo.  19  da. 


KATIONAL  ARITHMETIC  89 

348.    Problem  :    In  what  time  will  $560  amount  to 
$625.71  at  6%  interest? 

$625.71  $560 

560.  .06 


$ 

65.71    $33.60 

1  9556 

33.60)65.71 10000 

33  60 

32  11  0 

30  24  0 

187  00 

1  68  00 

19  000 

16  800 

2  2000 

2  0160 

First  find  the  interest  by 
deducting  the  principal,  $560, 
from  the  amount,  $625.71.  This 
shows  that  the  interest  is  $65.71. 
The  interest  at  6%  on  $560 
for  one  year  is  $33.60. 

Proceeding  as  in  the  previous 

illustrated  problem,  we  find  the 

time     required     to     be     1.9556 

years.        Reduced      to      years, 

months,  and  days,  this   equals 

1840  1  year  11  months  and  14  days 

1.9556  yr.  (par.  231). 

I     12 


11.4672  mo. 

I     30 
14.0160  da. 

Ans.  1  yr.  11  mo.  14  da. 

Note.     For  practice  problems  in  finding  the  time  see  par.  57. 

349.    To  Find  the  Rate:    Divide  the  interest  by  the 
interest  on  the  principal  for  the  given  time  at  1%. 

The  divisor  should  be  carried  to  the  fourth  decimal  place  if 
necessarv. 


^0 


RATIONAL  ARITHMETIC 


ILLUSTRATED   SOLUTIONS 

350.    Problem  :  At  what  rate  will  $475  earn  $8.84  in 
134  days  ? 


$475. 


4.75 


9.50 
.95 
^   .1583 
6)10.6083 

$1.7680 


134  da. 

Int.  for  60  da.  at  u/o 
Int.  for  120  da.  at  6% 
Int.  for  12  da.  at  6% 
Int.  for  2  da.  at  6% 
Int.  for  134  da.  at  6% 
Int.  for  134  da.  at  1% 


1.768)8.840 
8.840 

j^lXS ,    O  /q. 


The  interest  on  $475  for  134  days  at  1% 
is  $1.7680.  The  interest  at  1%  is  contained  in 
the  given  interest  5  times ;  therefore  the  inter- 
est must  be  reckoned  at  5%  in  order  to  produce 
$8.84  in  134  days. 


Note.     For  practice  problems  in  finding  the  rate  see  par.  58. 

351.  To  Find  the  Princijpal:  Divide  the  Given 
Interest  by  the  hiterest  on  one  dollar  for  the  given 
time  at  the  given  rate. 

352.  Divide  the  Given  Amount  bv  the  Amount  of 
one  dollar  for  the  given  time  at  the  given  rate. 


The    divisor   must  be  absolutely  correct.     The  interest  on   $1 
should  be  carried  to  three  places  and  all  fractions  must  he  retained. 


RATIONAL  ARITHMETIC  91 

ILLUSTRATED   SOLUTIONS 

353.    Problem  :    What  principal  will  be  required  to 
earn  $152.50  in  1  year  7  months  20  days  at  8%  ? 

1. 


.01  Int.  for  2  mo.  at  6% 

.06  Int.  for  1  yr.  at  6% 

.03  Int.  for  6  mo.  at  6% 

.005  Int.  for  1  mo.  at  6% 

.003i  Int.  for  20  da.  at  6% 

3).098i  Int.  for  1  yr.  7  mo.  20  da.  at  K^yo 

.032J-  Int.  for  1  yr.  7  mo.  20  da.  at  2% 

$.131i  Int.  for  1  yr.  7  mo.  20  da.  at  8% 


.13H)152.50 

$11  63|135 

1.180)1372.5O|OOO 
118 


74  5 

70  8 


IQQ  First  find  the  interest  on  $1   for  1 

1 1  o  year   7   months  and  20  days  at  8%  by 

the    sixty-day    method,     retaining    all 
fractions  (par.  333).     This  is  $.131^. 
If    $1    produces    $.131^    in    1    year 
^'^^  7  months  and  20  days  at  8%,  it  will 

•^  ^^  take  as  many  dollars  to  produce  $152.50 

16  0  as     $.131i     is     contained    in    $152.50. 

11  8  Performing  this  division  (par.  224),  we 

4  20         find    that    $1163.14    is    the    principal 
3  54         required. 

660 
590 


70 

Arts.  $1163.14. 


92 


RATIONAL  ARITHMETIC 


354.    Problem  :  What  principal  will  amount  to  $1250 
in  287  days  at  6%  ? 


1. 


.01 

.04 

.005 

.002i 

.000^ 


1.047 


Int.  for  60  da.  at  \t-/o 
Int.  for  240  da.  at  6% 
Int.  for  30  da.  at  6% 
Int.  for  15  da.  at  6% 
Int.  for  2  da.  at  6% 
Amount  on  $1  for  287  da.  at 


0 


1.0471)1250. 


$1  192  937 


6.287)7500.000  000 

6287 


1213  0 

628  7 

584  30 

565  83 

18  470 

12  574 

5  896  0 

5  658  3 

237  70 

188  61 

49  090 

44  009 

First  find  the  interest  on  $1  for 
287  days  at  6%.  This  is  $.047f. 
Add  this  to  $1  and  we  find  that  $1 
will  amount  to  $1.047f. 

It  will  take  as  many  dollars  to 
amount  to  $1250  as  $1.047f  is  con- 
tained in  $1250,  which  is  $1192.94. 


5  081 


Ans.  $1192.94. 


RATIONAL  ARITHMETIC  93 

355.  The  work  of  finding  the  interest  on  $1  at  6% 
may  be  simpHfied  by  using  the  following  rule : 

Multiply  the  number  of  years  by  6.  Call  the  result 
cents. 

Divide  the  number  of  months  by  2.  Call  the  result 
cents. 

Divide  the  number  of  days  by  6.  Call  the  result 
mills,  or  tenths  of  a  cent. 

356.  Applying  this  rule  in  the  above  illustrated 
solutions  we  would  have : 

In  the  first :  Find  the  interest  on  $1  for  1  year  7 
months  20  days  at  6%,  thus : 

6X1  =  6    written  .06 
7 -f- Q  =  3i  written  .035 
20  -^  6  -  3i  written  .0331 

.128^  =  Int.  on  $1  at  6% 

In  the  second  :  To  find  the  interest  on  $1  for  287  days 
at  6%,  thus  : 

287-^6  =  471  written  .0471  =  Int.  on  $1  at  6% 


Note.     For  practice  problems  in  finding  the  principal  see  par.  59. 
For  general  problems  in  interest  see  par.  60. 


COMMERCIAL   PAPERS 

357.  Commercial  Papers  comprise  notes,  drafts,  and 
checks. 

358.  A  Note  is  a  written  promise  of  one  party  to 
pay  a  second  party  a  certain  sum  at  a  certain  time. 

There  are  two  parties  to  a  note. 

359.  The  Maker  is  the  party  who  promises  to  pay. 

360.  The  Payee  is  the  party  to  whom,  or  to  whose 
order,  payment  is  to  be  made. 

361.  A  Draft  is  a  written  order  of  one  party  telling 
a  second  party  to  pay  a  third  party  a  certain  sum  at  a 
certain  time. 

There  are  three  parties  to  a  draft. 

362.  The  Drawer  is  the  party  requesting  payment. 

363.  The  Drawee  is  the  party  to  whom  this  request 
is  addressed  :   the  party  told  to  pay. 

364.  The  Payee  is  the  party  to  whom,  or  to  whose 
order,  payment  is  to  be  made. 

365.  A  Sight  Draft  is  a  draft  which  by  its  terms  is 
to  be  paid  by  the  drawee  immediately  upon  its  presen- 
tation to  him. 

(a)  In  some  states,  three  days,  called  days  of  grace,  are  allowed 
on  sight  drafts. 

94 


RATIONAL  ARITHMETIC  95 

366.  A  Time  Draft  is  a  draft  payable  at  a  certain 
time  after  presentation  to  the  drawee,  or  after  date. 

367.  The  Date  of  Maturity  of  either  a  note  or  a 
draft  is  the  date  upon  which  the  payment  is  due. 

(a)  To  fix  the  maturity  of  a  time  draft  due  "after  sight,"  it  is 
necessary  to  present  the  draft  to  the  drawee  to  see  if  he  is  wiUing 
to  pay  it.  If  he  is  wiUing  to  pay  it,  he  so  indicates  by  writing  the 
word  "accepted,"  together  with  the  date,  over  his  signature,  across 
the  face. 

(b)  The  maturity  of  a  note  is  ascertained  by  reckoning  the 
specified  time  from  date  of  the  note. 

(c)  The  maturity  of  a  draft  drawn  "after  sight"  is  ascertained 
by  reckoning  the  specified  time  from  the  date  of  acceptance. 

(d)  The  maturity  of  a  draft  drawn  "afterdate"  is  ascertained 
by  reckoning  the  specified  time  from  the  date  of  the  draft. 

368.  The  Face  of  a  note  or  draft  is  the  amount  of 
money  mentioned  in  it. 

369.  The  Amount  due  at  maturity  is  the  sum  that 
is  to  be  paid.  This  may  be  the  face,  or  it  may  be  the 
face  plus  interest. 

370.  A  Check  is  an  order  by  one  who  has  funds  on 
deposit  in  a  bank,  telling  the  bank  to  pay  a  certain 
sum  from  this  deposit  to  a  certain  party.  Checks  are 
treated  as  cash. 

(a)  The  drawer  of  a  check  is  the  depositor. 

(6)  The  drawee  of  a  check  is  the  bank  where  the  funds  are  on 
deposit. 

(c)  The  payee  of  a  check  is  the  party  in  whose  favor  the  check  is 
made :  The  party  to  whom  funds  from  the  deposit  are  to  be  paid 
by  the  bank. 


PARTIAL  PAYMENTS 

371.  It  is  sometimes  necessary  to  make  a  part  pay- 
ment on  a  promissory  note  or  other  obligation.  It  is 
usually  customary  in  such  cases  to  cancel  the  original 
note  and  issue  another  for  the  reduced  amount.  When 
it  is  not  feasible  to  do  this,  the  amount  of  the  part 
payment,  together  with  the  date,  is  indorsed  on  the 
back  of  the  original  instrument. 

Various  rules  are  in  use  for  finding  the  balance  due 
on  obligations  upon  which  part  payments  have  been 
indorsed. 

The  more  important  of  these  are  the  United  States 
Rule  and  the  Merchants'  Rule.  The  United  States 
Rule  has  been  sanctioned  by  the  Supreme  Court  of 
the  United  States.  The  Merchants'  Rule  is  used  by 
most  bankers  and  business  men  because  of  its  sim- 
plicity. 

THE   UNITED    STATES   RULE   FOR   PARTIAL   PAYMENTS 

372.  To  Find  the  Balance  Due  at  a  Given  Time: 
Find  the  interest  on  the  principal  from  the  date  of 
the  instrument  to  the  date  of  the  first  payment.  If 
this  interest  is  less  than  the  first  payment,  add  the 
interest  to  the  original  principal  and  subtract  the 
payment  from  this  amount.     Treat  the  remainder  as 

96 


RATIONAL  ARITHMETIC  97 

a  new  principal  and  proceed  as  before,  so  continuing 
until  the  date  of  settlement  is  reached. 

If  at  any  time  the  interest  is  greater  than  the  pay- 
ment, the  interest  should  be  disregarded  and  the 
interest  found  to  such  date  as  the  sum  of  the  payments 
exceeds  the  interest. 

ILLUSTRATED   SOLUTION 

373.  Problem  :  What  is  the  balance  due  July  1,  1916, 
on  a  note  of  $1200,  dated  January  1,  1914,  upon  which 
the  following  payments  have  been  made :  June  24, 
1914,  $250  ;  August  16,  1914,  $100  ;  July  8,  1915,  $40  ; 
January  1,  1916,  $300.^ 

$1200  Face  Jan.  1,  1914 

34.60  Int.  to  June  24,  1914,  5  mo.  23  da. 
$1234.60 

250.  1st  payment 

984.60  New  principal  June  24,  1914 

8.53  Int.  to  Aug.  16,  1914,  1  mo.  22  da. 
993.13 

100.  2d  payment 

893.13  New  principal  Aug.  16,  1914 

73.68  Int.  to  Jan.  1,  1916,  1  yr.  4  mo.  15  da. 
(Third  payment  not  equal  to  interest) 
966.81 

340.  3d  and  4th  payments 

626.81  New  principal  Jan.  1,  1916 

18.80  Int.  to  July  1,  1916,  6  mo. 
$645.61 

Ans,  $645.61. 


98  RATIONAL  ARITHMETIC 

$1200  draws  interest  from  January  1,  1914,  to  June  24,  1914. 
We  find  this  time  (par.  238)  to  be  5  months  and  23  days.  The 
interest  on  $1200  for  5  months  and  23  days  is  $34.60  (par.  335), 
making  the  amount  due  on  June  24,  1914,  $1234.60,  upon  which 
$250  is  paid,  leaving  the  balance  of  $984.60.  This  is  on  interest 
from  June  24,  1914,  to  August  16,  1914,  a  period  of  1  month  and 
22  days.  The  interest  on  $984.60  for  1  month  and  22  days  is 
$8.53,  making  $993.13  due  August  16,  1914.  On  this  $100  is  paid, 
leaving  a  balance  of  $893.13  to  draw  interest  from  August  16,  1914. 
The  next  payment  is  made  on  July  8,  1915.  The  interest  on 
$893.13  from  August  16,  1914,  to  July  8,  1915,  is  greater  than 
the  payment ;  therefore  we  disregard  the  interest  at  this  time  and 
find  the  interest  to  the  date  of  the  next  payment,  January  1,  1916. 
The  time  from  August  16,  1914,  to  January  1,  1916,  is  1  year  4 
months  and  15  days.  The  interest  on  $893.13  for  1  year  4  months 
and  15  days  is  $73.68,  making  $966.81  due.  On  this  $40  was  paid 
on  July  6,  1915,  and  $300  on  January  1,  1916,  which  is  $340  in  all, 
leaving  $626.81  to  draw  interest  from  January  1,  1916,  to  the  date 
of  settlement,  July  1,  1916,  6  months.  The  interest  on  $626.81  for 
6  months  is  $18.80,  making  $645.61  due  on  July  1,  1916. 

Note.     For  practice  problems  in  partial  payments  (United  States  Rule) 
see  par.  61. 

MERCHANTS'   RULE 

374.  To  Find  the  Balance  Due  at  a  Given  Time: 
Find  the  interest  on  the  face  of  the  obligation  from  the 
date  at  which  it  begins  to  draw  interest  to  the  date  of 
final  settlement.  Add  this  interest  to  the  face  of  the 
debt.  Find  the  interest  on  each  payment  from  the 
date  of  the  payment  to  the  date  of  final  settlement. 
Add  the  payments  and  the  interest  on  the  payments. 
Subtract  this  sum  from  the  amount  of  the  principal 
and  interest.  The  difference  will  be  the  balance 
due. 


RATIONAL  ARITHMETIC  99 

ILLUSTRATED  SOLUTION 
375.  Problem  :  What  is  the  balance  due  Julv  1, 
1916,  on  a  note  of  $1200,  dated  January  1,  1914,  upon 
which  the  following  payments  have  been  made :  June 
24,  1914,  $250;  August  16,  1914,  $100;  July  8,  1915, 
$40;   January  1,  1916,  $300? 

$1200     Face  Jan.  1,  1914 

180     Int.  to  July  1,  1916,  2  yr.  6  mo. 
$1380 
$250.         Paid  June  24,  1914 

30.29     Int.  to  July  1,  1916,  2  yr.  7  da. 
100.         Paid  Aug.  16,  1914 
11.25     Int.  to  July  1,  1916,  1  yr.  10  mo.  15  da. 
40.         Paid  Julv  8,  1915 
2.35     Int.  to  July  1,  1916,  11  mo.  23  da. 
300.         Paid  Jan.  1,  1916 

9.         Int.  to  Julv  1,  1916,  6  mo. 


$742.89 

$1380 
742.89 


$637.11  Balance  due  July  1,  1919. 

A  $1200  note  given  January  1,  1914,  would  earn  $180  interest 
to  July  1,  1916,  making  its  value  at  maturitj^  $1380.  $250  paid 
June  24,  1914,  would  earn  $30.29,  interest  to  July  1,  1916,  a  period 
of  2  years  7  days.  $100  paid  August  16,  1914,  would  earn  $11.25 
to  July  1,  1916.  $40  paid  on  July  8,  1915,  would  accrue  $2.35  in- 
terest to  July  16,  1916.  $300  would  accrue  $9  interest  to  July  1, 
1916.  The  payments  and  accrued  interest  amount  to  $742.89. 
The  value  of  the  note  at  maturity,  $1380,  minus  the  payments 
and  accrued  interest  $742.89,  leaves  $637.11  due  on  July  1,  1916. 

Note.  For  practice  problems  in  partial  payments  (Merchants'  Rule) 
see  par.  62. 


BANK  DISCOUNT 

376.  Bank  Discount  is  the  charge  made  by  a  bank 
for  cashing  an  obHgation  before  it  is  legally  due.  It  is 
the  interest  on  the  amount  due  at  maturity  for  the 
unexpired  time. 

377.  The  Maturity  of  a  debt  is  the  date  upon  which 
it  becomes  legally  due. 

{a)  A  few  states  allow  three  days  in  addition  to  the  time  men- 
tioned in  a  note  or  draft.     Tliese  are  called  days  of  grace. 
{b)  Most  states  allow  days  of  grace  on  sight  drafts  only. 

378.  The  Term  of  Discount  is  the  number  of  days 
between  the  date  of  discount  and  the  date  of  maturity. 

379.  The  Bank  Discount  is  the  interest  on  the 
amount  due  at  maturity  for  the  term  of  discount. 

380.  The  Proceeds  is  the  difference  between  the 
amount  due  at  maturity  and  the  bank  discount.  It 
is  the  cash  value  of  the  debt  on  the  date  of  discount. 

381.  To  Find  the  Proceeds :  Find  the  date  of 
maturity.  Ascertain  the  amount  due  at  maturity. 
Find  the  time  in  exact  days  from  the  date  of  discount 
to  the  date  of  maturity.  Compute  the  interest  on  the 
amount  due  at  maturity  for  this  time.  The  result  will 
be  the  bank  discount.  Deduct  the  bank  discount 
from  the  amount  due  at  maturity.  The  result  will  be 
the  proceeds. 

100 


RATIONAL  ARITHMETIC    '   ^  ^         X5l      -  ,'  :> 

ILLUSTRATED   SOLUTIONS        ,  >'^  '■>      '->  >'\}l^}  i\5  >  .V 

382.  Problem  :  Find  the  bank  discount  and  the 
proceeds  of  a  note  for  60  days  for  $5000,  dated  March  3, 
1916,  discounted  April  1,  1916,  at  5%.  | 

March    3,   1916  +  60    days  =  May  2,   1916  =  date    of  i 

maturity.  i 

April    1,  1916,  to   May  2,  1916  =  31  days  =  term  of 

discount  ' 

A  note  given  March  3, 

$5000.  1916,  for  60  days  would 

50.  fall  due  on  May  2,  1916, 

25  30  days  which  would  be  the  date 

~  '8333     Idav  f  '"T'f,  ^,t/™' 

- — —  "  from    April    1,    1916,    to 

6)25.8333     6^  jy^^^  2^  l9jg^  j^  3j  ^^^.^ 

4.30oo      1%  This  is  the  term  of  dis- 

21.5278      5%  Bank  Discount     count.     The    interest    on 

$5000  for  31  days  at  5% 

$5000.  is    $21.53.     This    is    the 

21  53  bank  discount.     The  face 

$4978.47     Proceeds  °f   the   note   was   $5000, 

the  bank  discount  $21.53  ; 
Ans.    $4978.47.  the  net  proceeds  would  be  i 

the  difference,  $4978.47. 

Note.     For  practice  problems  in  finding  the  bank  discount  and  proceeds 
of  non-interest-bearing  notes  see  par.  63. 

I 

383.    Problem  :  Find  the  bank  discount  and  proceeds 
of  a  sixty-day  note  for  $5000,  dated  January  1,  1915,  . 

bearing  interest  at  6%,  discounted  February  6,  1915, 
at-  o  /q. 

January  1,  1915  +  60  days  =  March  2,  1915  j 

February  6,  1915,  to  March  2,  1915  =  24  days 


$5000     Face 

50     Interest 

$5050     Due 

at  Maturity 

50.50 

60  days 

6)20.20 

24  days  at  6% 

3.3666 

$16.8334 

Bank  Discoun 

$5050 

16.83 

■> 

102  RATIONAL  ARITHMETIC 

A  note,  dated  January  1, 
1915,  to  run  60  days  would 
fall  due  on  March  2,  1915. 
If  it  were  discounted  on 
February  6,  1915,  it  would 
then  have  24  days  to  run. 
The  term  of  discount,  there- 
fore, is  24  days.  xA.s  this 
note  is  given  with  interest,  its 
face  value,  plus  60  days'  inter- 
est on  $5000,  is  $5050.  As 
the  amount  due  at  maturity 

^5033  17      Proceeds  ^^  $5050,  the  bank  discount 

would    be    figured    on    this 
Ans.   $5033.17.  amount  for  24  days  at  5%, 

which  is  $16.83.  The 
amount  due  at  maturity  being  $5050  and  the  bank  discount 
$16.83,   the  net  proceeds  would  be  the  difference,   or  $5033.17. 

Note.     For  practice  problems  in  ifinding  the  bank  discount  and  proceeds 
of  interest-bearing  notes  see  par.  64. 


384.  In  making  a  loan  at  a  bank  when  a  definite 
amount  is  desired,  the  note  must  be  made  for  a  sum 
that,  when  discounted,  will  leave  as  the  net  proceeds 
the  amount  of  loan  desired. 

To  Find  the  Sum  for  Which  a  Note  Must  Be  Drawn 
So  That,  if  Discounted  at  Date,  the  Proceeds  Will  Be  a 
Given  Sum :  Find  the  proceeds  of  a  note  for  $1  for 
the  given  time  at  the  given  rate.  Divide  the  given 
proceeds  by  this.  The  quotient  will  be  the  face 
required. 

The  divisor  must  be  absolutely  correct.  Carry  the  discount  on 
$1  to  the  third  place  and  retain  all  fractions. 


RATIONAL   ARITHMETIC  103 

ILLUSTRATED   SOLUTION 

385.  Problem  :  For  what  sum  must  a  ninety-day 
note  be  drawn  so  that  if  discounted  on  its  date  at  4^% 
the  proceeds  may  be  $1875  ? 

$1.00 

.01         60  days 
.005       30  days 


4).015       6% 
.0031     li% 
.01  li     Bank  Discount  on  $1 

.9881     Proceeds  on  $1 

$1875 --.9881 


1  896  333 

3.955)7500.000  000 
3955 

3545  0 
3164  0 

38100 
355  95 

25  050 
23  730 

1  320  0 
1  186  5 

133  50 
118  65 

14  850 
11865 

The  bank  discount  on  a  note  of  $1  for 
90  days  at  ^%  would  be  $.01 1^.  The 
net  proceeds  of  $1  would  be  $.988f. 
If  $1  yields  proceeds  of  $.988f,  it 
would  take  as  many  dollars  to  yield 
$1875  as  $.988f  is  contained  in  $1875, 
which  is  1896.33,  the  face  of  the  note 
required. 


2  985       Ans,     $1896.33. 

Note.  For  practice  problems  in  finding  the  sum  for  which  a  note  must 
be  drawn  so  that  if  discounted  at  date  the  proceeds  will  be  a  given  sum,  see 
par.  65. 


COMPOUND   INTEREST 

386.  Compound  Interest  is  interest  on  the  principal 
and  interest  combined,  as  fast  as  the  interest  falls 
due. 

(a)  Compound  interest  can  only  be  charged  by  special  agreement, 
and  then  care  must  be  exercised  that  the  laws  of  usury  are  not 
violated. 

{b)  Interest  is  usually  compounded  annually,  semi-annually,  or 
quarterly. 

(c)  Compound  interest  is  little  used  except  in  savings  banks. 

387.  To  Find  Compound  Interest :  Find  the  amount 
of  the  principal  and  interest  at  the  end  of  the  first 
interest  period.  Use  this  amount  as  a  new  principal 
for  the  next  period,  and  so  on.  Deduct  the  original 
principal  from  the  final  amount.  The  difference  will 
be  the  compound  interest. 

ILLUSTRATED   SOLUTION 

388.  Problem  :  Find  the  compound  interest  on 
$1400  from  March  8,  1912,  to  June  15,  1916,  at  5%, 
interest  compounded  annually. 

1916     6     15 
1912     3       8 

4     3       7  =  Four  full  periods  and  3  mo.  7  da.  extra 

104 


RATIONAL  ARITHMETIC 


105 


$1400 

70 

$1470 

73.50 
$1543.50 
77.18 
$1620.68 
81.03 
$1701.71 
17.0171 
17.0171 
8.5085 
1.7017 
.2836 


6)27.5109 

4.5851 

22.9258 

$1724.6358 

1400 


Original  Principal 

Int.  at  5%  for  1st  period 

New  Principal 

Int.  at  5%  for  2d  period 

New  Principal 

Int.  at  5%  for  3d  period 

New  Principal 

Int.  at  5%  for  4th  period 

New  Principal 

Int.  for  2  mo.  at 

Int.  for  2  mo.  at 

Int.  for  1  mo.  at 

Int.  for  6  da.  at 

Int.  for  1  da.  at  u/o 

Int.  for  3  mo.  7  da.  at  u/o 

Int.  for  3  mo.  7  da.  at  1% 

Int.  for  3  mo.  7  da.  at  5% 

Final  Amount 


0 


$324.6358     Compound  Int.     Ans.  $324.64. 

Note.     For  practice  problems  in  compound  interest  see  par.  66. 


PERIODIC   OR   ANNUAL   INTEREST 

389.  Periodic  Interest,  often  called  annual  interest, 
is  interest  on  the  principal  and  interest  on  each  overdue 
payment  of  interest. 

(a)  It  is  the  result  of  a  business  custom  in  certain  lines,  and  is 
merely  an  application  of  the  general  principles  of  simple  interest. 

(6)  It  is  not  entitled  to  be  considered  as  a  separate  arithmetical 
subdivision,  and  is  only  so  treated  in  this  book  because  many  authors 
have  seen  fit  to  introduce  it  as  a  separate  kind  of  interest. 

(c)  It  has  no  legal  status. 


106  RATIONAL  ARITHMETIC 

ILLUSTRATED   SOLUTIONS 

390.  Problem  :  $1400  was  loaned  on  January  8, 
1915,  for  2  years  at  6%,  interest  payable  semi-annually ; 
each  installment  of  interest  to  draw  interest  at  6% 
from  its  due  date  until  paid.  What  sum  would  be 
required  to  cancel  the  debt  and  all  interest  on  January  8, 
1917,  nothing  having  been  paid  previously  ? 

$1400. 

14.     Int.  for  2  mo.  at  6% 


$42.     Int.  for  6  mo.  at  6% 
4. 


$168.     Int.  for  2  yr.  at  6% 

1  yr.       6  mo.  —  1st  installment  overdue 

1  yr.  — 2d 
6  mo.  — 3d 

2  yr.     12  mo.  =  3  yr. 

$42. 


'0 
0 


.42  Int.  on  interest  for  2  mo.  at  6% 

2.52  Int.  on  interest  for  1  yr.  at 

5.04  Int.  on  interest  for  2  yr.  at 

7.56  Int.  on  interest  for  3  yr.  at 

168.  Int.  on  the  Principal 

1400.  Principal 

$1575.56  Amount  due  Jan.  8,  1917 

Ans.  $1575.56. 

The  interest  on  $1400  for  6  months  at  6%  is  $42.  Then  $42 
should  be  paid  every  6  months.  There  would  be  four  such  pay- 
ments due  in  2  years.     4  X42  =  $168,  which  must  be  paid  as  interest 


RATIONAL  ARITHMETIC  107 

on  the  principal.  The  first  installment  of  this,  amounting  to  $42, 
was  due  on  July  8,  1915.  Tliis  is  overdue  1  year  6  months.  The 
next  installment  of  $42  is  overdue  1  year ;  the  next,  6  months ; 
and  the  last  is  just  due.  Besides  the  interest  on  the  principal, 
then,  there  is  interest  on  the  interest  due.  This  last  is  the  interest 
on  $42  for  1  year  6  months,  and  for  1  year,  and  for  6  months.  In 
other  words,  the  interest  on  the  interest  equals  the  interest  on  $42 
for  3  years,  w^iich  is  $7.56.  Adding  interest  on  interest,  interest 
on  principal,  and  principal  gives  $1575.56,  the  amount  due  at 
maturity. 

Note.     For  practice  problems  in  periodic  interest  see  par.  67. 


AVERAGING   ACCOUNTS 

391.  Averaging  of  Accounts  is  the  process  of  ascer- 
taining the  date  on  which  an  account  may  be  paid 
without  loss  of  interest  to  either  the  debtor  or  the 
creditor.  ^ 

Averaging  of  accounts  has  been  abandoned  by  most  lines  of  busi- 
ness. It  is  now  the  general  custom  to  settle  each  item,  separately, 
at  maturity.     The  subject,  however,  is  not  entirely  obsolete. 

392.  Cash  Balance  is  the  amount  of  cash  required 
to  settle  an  account  without  loss  of  interest  to  either 
party  on  any  date  other  than  the  average  due  date. 

GENERAL   PRINCIPLES    OF   AVERAGE 

393.  If  an  account  is  paid  before  it  is  due,  the  payer 
loses  the  interest  on  the  sum  paid,  and  the  receiver 
gains  it. 

394.  If  an  account  is  paid  after  it  is  due,  the  payer 
gains  the  interest  on  the  money  paid  and  the  receiver 
loses  it. 

395.  The  average  due  date  of  several  items,  due  at 
different  dates,  is  the  date  when  the  payer's  losses  of 
interest  and  gains  of  interest  would  be  equal,  or  within, 
one  half  day's  interest  of  being  equal. 

108 


RATIONAL  ARITHMETIC 


109 


396.  To  Average  an  Account:  Assume  a  date  of 
settlement.  Find  the  net  gain  or  loss  of  interest  by 
paying  on  that  date.  Find  how  long  it  will  take  the 
amount  to  be  paid  to  earn  this  interest.  Count  that 
time  forward  or  backward  from  the  assumed  date 
according  to  whether  the  payer  would  lose  or  gain. 

(a)  The  assumed  date  Is  called  the  focal  date. 

{h)  The  focal  date  may  be  any  date. 

(c)  The  easiest  focal  date  to  use  is  the  zero  date  of  the  earliest 
month  in  which  any  one  of  the  items  is  due ;  thus,  if  items  are  due 
June  8,  July  15,  and  August  9,  the  easiest  focal  date  to  use  would 
be  June  0,  which  is,  in  reality.  May  31. 

ILLUSTRATED   SOLUTIONS 

397.  Problem  :  Average  the  follow^ing  : 

Charles  S.  Chase 


1916 

Jan.  5 

$434. 

27 

123.50 

Feb.  8 

215. 

Apr.  9 

310.65 

Focal  date  Jan.  0. 


Jan. 

5 

434. 

5  da. 

.3616 

27 

123.50 

27  da 

.5557 

Feb. 

8 

215. 

39  da. 

1.3975 

Apr. 

9 

310.65 

99  da. 

5.1256 

1083.15 


7.4404 


110  RATIONAL  ARITHMETIC 

41 


6)1.0831     Int.  for  6  da.  .1805)7.4404 

.1805     Int.  for  1  da.  7  220 


2204 

1805 

399 


Ans.  Jan.  0+41=  Feb.  10. 


Assuming  Januarj^  0  as  the  focal  date,  on  the  first  item,  $434, 
Chase  would  lose  5  days'  interest,  because  he  would  pay  it  5  days 
before  it  was  due.  The  interest  on  $434  for  5  days  is  $.3616.  By 
paying  $123.50  on  January  0,  the  interest  for  27  days  would  be 
lost,  which  would  be  $.5557.  By  paying  $215,  due  February  8,  on 
January  0,  the  interest  for  39  days  would  be  lost,  which  would  equal 
$1.3975.  If  $310.65  is  paid  99  days  before  it  is  due,  the  interest 
lost  would  be  $5.1256.  Then  by  paying  $1083.15  on  January  0, 
$7.4404  interest  would  be  lost.  The  interest  on  $1083.15  for  one 
day  is  $.1805.  It  will  take  as  many  days  to  earn  $7.4404  as 
$.1805  is  contained  in  $7.4401,  which  is  41  days.  Then  Chase 
should  pay  the  money  41  days  later  than  January  0,  which  is 
February  10. 

Note.     For  practice  problems  in  averaging  accounts  see  par.  68. 


398.    Problem  :   Average  the  following  : 
Find  cash  balance  on  Jan.  1,  1916. 


1915 

Sept.    4 

$625 

for    2  mo. 

Credit 

Oct.    23 

350 

for  30  da. 

Credit 

Nov.  18 

215 

for  10  da. 

Credit 

Dec.     8 

643 

for  60  da. 

Credit 

RATIONAL   ARITHMETIC  111 


Sept. 

4 

2  mo. 

Nov. 

4 

$  625 

4  da. 

$  .4166 

Oct. 

23 

30  da. 

Nov. 

22 

350 

22  da. 

1.2832 

Nov. 

18 

10  da. 

Nov. 

28 

215 

28  da. 

1.0032 

Dec. 

8 

60  da. 

Feb. 

6 

643 

98  da. 

10.5023 

Focal  date  Nov.  0. 


$1833  $13.2053 

43 


)1.833 

Int.  for  6  da. 

.3055)13.2053 

.3055 

Int.  for  1  da. 

12  220 
9853 
9165 

688 
Nov.  0  +  43  da.  =  Dec.  13. 

Dec.  18  1833.  19  days  at  6% 

Jan.  J.  18.33         Int.  for  60  da.  at  6% 

19  days  4.5825     Int.  for  15  da.  at  6% 


.9165  Int.  for  3  da.  at  6% 

.3055  Int.  for  1  da.  at  6% 

5.8045  Int.  for  19  da.  at  6% 
1833 


$1838.80         Cash  Bal.  Jan.  1,  1916 

.  f  Dec.  13,  1915. 

^^^'    I  $1838.80. 

Goods  billed  on  September  4  for  2  months'  credit  would  be  due 
on  November  4.  A  bill  bought  October  23  on  30  days'  credit 
would  be  due  November  22.  A  bill  bought  November  18  on  10 
days'  credit  would  be  due  November  28.  A  bill  bought  December  8 
on  60  days'  credit  would  be  due  on  February  6,  Assuming  Novem- 
ber 0  as  the  focal  date  and  averaging  as  in  par.  397,  we  find  the 
average  due  date  to  be  December  13,  1915. 

If  the  debtor  should  pay  $1833  on  December  13,  but  did  not  do 


112  RATIONAL  ARITHMETIC 

so  until  January  1,  he  would  owe  $1833  plus  19  days'  interest.  The 
interest  on  $1833  for  19  days  is  $5.8045,  making  the  amount  due 
January  1,  $1838.80. 

Note.     For  practice  problems  in  averaging  accounts  and  finding  the 
cash  balance  see  par.  69. 

399.    Problem  : 
Dr.  C.  E.  Batchelor  Cr. 


1915 

1915 

May  8 

30 

days 

$525.30 

May 

15 

Cash 

$300. 

June  7 

60 

days 

415.40 

June 

5 

Cash 

425.50 

When  is  the  above  due  by  average  ? 

What  was  the  cash  balance  December  15,  1915  .^^ 

May  8+30  da.  =  June  7     $525.30     38     $3.3269 
June  7  +  60  da.  =  Aug.  6       415.40     98       6.7848 


940.70 

10.1117 

725.50 

3.303 

$215.20 

$6.8087 

May  15  $300 

15  $.75 

June  5   425.50 

36  2.553 

$725.50  $3,303 

Focal  date  May  0. 


190 


6).2152     Int.  for  6  da.  at  6%         .0358)6.8087 

.0358     Int.  for  1  da.  at  6%  3  58 

3  228 
3  222 

67 


RATIONAL  ARITHMETIC  113 

May  0  +  190  days  =  Nov.  6,  1915. 
Nov.  6  to  Dec.  15  is  39  days. 

$^15.20 

2.15  Int.  for  60  da.  at  6% 

1.076  Int.  for  30  da.  at  6% 

.2152  Int.  for  6  da.  at  6% 

.1076  Int.  for  3  da.  at  6% 

1.3988  Int.  for  39  da.  at  6% 

215.20 

$216.60  Cash  Bal.  Dec.  15,  1915 


Ans. 


Nov.  6,  1915. 
$216.60. 


In  the  above  account  C.  E.  Batchelor  owes  the  items  on  the 
debit  side  of  the  account  and  we  owe  him,  theoretically,  the  items 
on  the  credit  side.  The  debit  side  being  the  larger,  he  owes  us  the 
balance  of  the  account.  First  find  the  due  dates  of  all  the  items. 
On  the  debit  side  $5'25. 30  is  due  June  7.  $415.40  is  due  August  6. 
On  the  credit  side  $300  is  due  May  15.  $4*25.50  was  due  June  5. 
Assuming  the  zero  date  of  the  earliest  month  as  the  focal  date,  this 
w^ould  be  May  0.  Batchelor,  by  paying  $525.30  on  May  0,  would 
lose  38  days'  interest,  which  is  $3.3269,  and  by  paying  $415,40  on 
May  0  he  would  lose  98  daj's'  interest,  which  is  $6.7848.  If  the 
account  were  settled  on  May  0,  Batclielor  would  gain  the  interest 
on  $300  for  15  days,  which  is  $.75  and  he  would  gain  the  interest 
on  $425.50  for  36  days,  which  is  $2,553.  This  would  amount  to 
$3,303.  The  balance  of  the  account  is  $215.20.  If  Batchelor  paid 
this  on  May  0,  he  would  lose  $10.1117  and  gain  $3,303,  or  he  would 
make  a  net  loss  of  $6.8087.  The  interest  on  $215.20  for  one  day 
is  $.0358.  To  make  up  $6.8087,  it  would  take  as  many  days  as 
$.0358  is  contained  in  $6.8087,  which  is  190  times,  or  190  days. 
Since  by  settling  the  account  on  May  0  the  interest  for  190  days  is 
lost,  it  should  be  settled  190  days  later,  which  would  be  Novem- 
ber 6. 


114 


RATIONAL  ARITHMETIC 


Whenever  the  balance  of  interest  and  balance  of 
account  fall  on  the  same  side,  the  payer  will  lose  the 
interest  and  the  time  should  then  be  counted  forward. 


400.    Problem 


Dr. 


F.  H.  Bray 


Cr. 


1916 

1916 

June 

6 

2 

mo. 

$2200 

Aug. 

12 

Cash 

$  108 

Aug. 

12 

12 

da. 

1400 

Sept. 

o 

Cash 

2892 

Find  the  cash  balance  for  Sept.   12,   1916,   at  ytyc. 
Focal  date  August  0. 


Aug.    6 

24 


$2200 
1400 

$3600 
3000 

$  600 


6 
24 


2.20 
5.60 
7.80 


Aug.  12  $  108  12       .216 
Sept.    5     2892  36  17.352 


$3000 


17.568 

7.80 
9.768 


6). 600     Int.  for  6  da.  at  u/o 
.10       Int.  for  1  da.  at  6% 


97 


.10)9.76 
90 
76 
70 


6 

8 


68 


RATIONAL  ARITHMETIC  115 

97.6  days  =  98  days. 

98  days  counted  backward  from  Aug.  0=  April  24, 
Average  Date. 

April  24  to  Sept.  12  =  141  days. 

Balance  of  account 


600 

$600. 

6. 

14.10 

12.      -- 

=  120 

$614.10 

2.      = 

=   20 

.10  = 

=      1 

14.10  = 

=  Int.  for  141  da 

• 

Ans.  $614.10. 

The  above  problem  is  similar  to  399,  except  that  the  balance 
of  account  and  the  balance  of  interest  fall  on  opposite  sides.  This 
amount  shows  that  F.  H.  Bray  owes  the  balance  of  $600.  By  pay- 
ing this  balance  on  August  0  he  would  lose  the  interest  on  $2''200 
for  6  days  and  the  interest  on  $1400  for  24  days,  or  $7.80  on  both 
items,  because  he  would  be  paying  before  due.  However,  his  loss 
of  interest  on  the  items  on  the  debit  side  of  the  account  is  more 
than  offset  by  his  gain  of  interest  on  the  items  on  the  credit  side 
(the  interest  on  $108  for  12  days  and  the  interest  on  $2892  for  26 
days)  which  is  $17,568.  The  difference  between  $17,568  and  $7.80 
is  $9,768,  which  is  the  amount  of  interest  that  Bray  would  gain  by 
paying  the  balance  on  August  0.  The  balance,  $600,  requires  98 
days  to  earn  $9,768.  Then  Bray,  in  order  to  neither  gain  nor 
lose,  should  settle  the  account  98  days  before  iVugust  0,  which  is 
April  24. 

When  the  balance  of  account  and  the  balance  of 
interest  fall  on  opposite  sides,  count  the  time  backward 
from  the  focal  date. 

Note.  For  practice  problems  in  finding  the  amount  due  by  average  and 
the  cash  balance  of  two-sided  accounts  see  par.  70. 


TAXES 

401.  A  Tax  is  a  sum  of  money  levied  upon  a  citizen 
or  his  property  to  meet  the  expenses  of  maintaining 
the  Government. 

(a)  Taxes  are  levied  to  pay  the  expenses  of  the  city,  county, 
state,  and  United  States. 

(h)  The  first  three  are  levied  directly  upon  the  person,  property, 
and,  in  some  cases,  the  income  of  the  individual. 

(c)  In  case  of  the  United  States,  the  tax  is  levied  through  duties 
and  customs  and  by  a  tax  on  incomes. 

402.  A  Poll  Tax  is  a  tax  levied  on  the  person,  and 
in  most  states  is  assessed  upon  all  male  citizens  of 
20  years  of  age  or  more. 

403.  A  Property  Tax  is  a  tax  assessed  on  property, 
either  real  or  personal,  and  is  levied  upon  all  persons 
owning  taxable  property,  irrespective  of  age  or  sex. 

404.  An  Income  Tax  is  a  tax  upon  incomes,  and  is 
levied  alike  on  all  citizens  receiving  certain  incomes, 
regardless  of  age  or  sex. 

Income  taxes  are  levied  by  the  United  States  Government  and  by 
some  states,  in  accordance  with  laws  passed  by  Congress  or  State 
Legislatures. 

Note.     For  practice  problems  in  taxes  see  par.  71. 

116 


RATIONAL   ARITHMETIC  117 

DUTIES   AND    CUSTOMS 

405.  Duties  and  Customs  are  taxes  assessed  by  the 
United  States  Government  on  imported  merchandise. 

406.  An  Ad  Valorem  Duty  is  a  certain  percentage 
of  the  net  cost  (value)  of  the  importation. 

407.  A  Specific  Duty  is  a  specified  sum  levied  on 
each  article,  or  on  each  unit  of  measure,  regardless  of 
the  value. 

(a)  Ad  valorem  duties  are  not  computed  on  fractions  of  a  dollar. 
Cents  are  disregarded  for  less  than  50  and  are  considered  another 
dollar  for  more  than  50. 

(b)  Some  articles  are  subject  to  both  ad  valorem  and  specific 
duties. 

(c)  Specific  duties  are  not  computed  on  fractions  of  a  unit.  The 
long  ton,  or  2240  pounds,  is  used  in  computing  specific  duties. 

408.  A  Tariff  is  a  schedule  showing  the  different 
rates  of  duties  imposed  by  Congress  on  different  articles. 

409.  A  Free  List  is  a  schedule  of  articles  upon  which 
no  duties  are  to  be  levied. 

410.  A  Customhouse  is  a  branch  office  of  the 
Treasury  Department  of  the  United  States  Govern- 
ment. 

Customhouses  are  established  at  various  ports ;  each  custom- 
house has  jurisdiction  over  certain  territory. 

411.  A  Port  of  Entry  is  a  port  where  a  customhouse 
is  established. 

412.  All  ports,  whether  of  entry  or  otherwise,  are 
called  ports  of  delivery. 


118  RATIONAL  ARITHMETIC 

413.  The  Customhouse  Business  is  distributed 
among  three  departments. 

414.  The  Collector's  Office  takes  charge  of  entries 
and  papers,  issues  permits,  and  collects  the  duties. 

415.  The  Surveyor's  Office  takes  charge  of  the 
vessels  and  cargoes,  receives  the  permits,  ascertains 
the  quantities,  and  delivers  the  merchandise  to  the 
importer. 

416.  The  Appraiser's  Office  examines  the  mer- 
chandise and  determines  the  value  and  rate  of  duty 
on  the  goods. 

417.  Internal  Revenue  is  a  revenue  raised  by  the 
Government  by  placing  duties  on  such  articles  of 
luxury  as  may  be  determined  by  Congress.  These 
duties  are  collected  by  the  Treasury  Department  and 
vary  from  time  to  time  according  to  the  needs  of  the 
country. 

418.  A  Manifest  is  a  memorandum,  signed  by  the 
master  of  the  vessel,  showing  the  name  of  the  vessel, 
its  cargo,  and  the  names  and  addresses  of  the  con- 
signors and  consignees. 

419.  An  Invoice  is  a  detailed  statement  showing  the 
items  and  value  of  the  goods  imported  and  is  made 
out  in  the  weights  and  measures  of  the  country  of 
export. 

The  values  of  foreign  moneys  are  periodically  proclaimed 
by  the  Secretary  of  the  Treasury,  and  these  values  must  be  taken 
in  estimating  duties.     See  par.  468. 


RATIONAL  ARITHMETIC  119 

420.  A  Bonded  Warehouse  is  a  warehouse  provided 
for  the  storage  of  goods  upon  which  duties  have  not 
yet  been  paid. 

(a)  Any  importer  may  deposit  goods  in  the  warehouse  by 
giving  bond  for  the  payment  of  duties  on  the  goods  thus  stored. 

(6)  On  goods  remaining  in  bond  more  than  a  year,  10%  addi- 
tional duty  is  charged. 

(c)  Goods  left  in  a  government  warehouse  for  three  years  are 
forfeited  to  the  Government  and  sold  at  auction. 

(d)  Goods  may  be  withdrawn  from  a  warehouse  for  export 
without  payment  of  duty. 

421.  An  Excise  Duty  is  a  tax  levied  upon  goods  pro- 
duced and  consumed  in  the  United  States. 

In  this  class  come  taxes  upon  tobacco  and  such  articles  of 
luxury  as  Congress  may  from  time  to  time  prescribe. 

422.  If  goods  on  which  either  excise  or  import 
duties  have  been  paid  are  exported,  the  amount  of  duty 
is  refunded.     This  is  called  a  drawback. 

Note.     For  practice  problems  in  duties  and  customs  see  par.  72. 


INSURA.NCE 

• 

423.  Insurance  is  a  contract  bj^  which  one  party 
(the  insurer)  agrees  to  reimburse  another  party  (the 
insured)  in  case  of  damage  or  loss  to  the  latter's  prop- 
erty or  person. 

424.  The  insurance  business  is  usually  conducted 
by  corporations  called  Insurance  Companies,  which 
limit  their  operations  to  certain  classes  of  risks.  Some 
companies  handle  fire  insurance,  others  marine  in- 
surance, others  accident  insurance,  others  life  insurance, 
etc. 

425.  There  are  two  kinds  of  insurance  companies, 
stock  companies  and  mutual  companies. 

426.  A  Stock  Company  is  one  whose  capital  is  owned 
by  stockholders  who  share  the  profits  and  who  are 
liable  for  the  losses. 

427.  A  Mutual  Insurance  Company  is  one  in  which 
there  are  no  stockholders,  but  in  which  the  parties 
insured  share  the  profits  and  losses. 

428.  A  Policy  is  the  contract  of  insurance. 

429.  A  Premium  is  the  sum  paid  for  the  insurance. 
It  is  the  consideration. 

120 


RATIONAL  ARITHMETIC  121 

(a)  Sometimes  the  rate  is  expressed  as  a  certain  percentage 
of  the  value  insured  and  sometimes  at  so  much  per  hundred  dollars 
of  insurance. 

(b)  The  rate  of  premium  depends  upon  the  amount  and  the 
nature  of  the  risk  and  the  length  of  time  for  which  the  risk  is  taken. 

430.  Fire  Insurance  is  insurance  against  loss  or 
damage  by  fire,  or  from  the  means  employed  for  ex- 
tinguishing it,  or  to  prevent  its  spread. 

431.  Owners  of  property  may  insure  in  one  or  more 
companies.  When  the  risk  is  placed  in  several  com- 
panies, care  should  be  taken  to  have  the  policies 
uniform  in  every  particular.  Each  company  will  then 
pay  such  part  of  the  total  loss  as  its  risk  is  of  the  total 
risk, 

432.  The  Average  Clause,  contained  in  manj^  policies, 
is  to  the  effect  that  the  liability  of  the  company  in  case 
of  a  partial  loss  shall  be  such  part  of  the  loss  as  the  insured 
value  is  of  the  actual  value  of  the  property. 

Thus,  a  building  worth  $20,000  is  insured  for  $18,000,  which 
is  nine-tenths  or  90%  of  the  value.  In  case  of  loss  the  company 
would  pay,  under  the  average  clause,  90%  of  the  loss. 

433.  Short  Rate  is  the  rate  for  less  than  a  year. 

434.  Marine  Insurance  is  insurance  against  loss  or 
damage  to  a  vessel  or  her  cargo  by  storm  or  other 
dangers  of  the  sea. 

Marine  policies  always  contain  the  average  clause. 
Note.     For  practice  problems  in  insurance  see  par.  73. 


122  RATIONAL  ARITHMETIC 

LIFE   INSURANCE 

435.  Under  this  head  are  considered  life  insurance^ 
accident  insurance,  and  health  insurance. 

436.  Life  Insurance  is  indemnity  for  loss  of  life. 

437.  Accident  Insurance  is  indemnity  for  loss  or 
disability  caused  by  accident. 

438.  Health  Insurance  is  indemnity  for  loss  oc- 
casioned by  sickness. 

439.  There  are  two  kinds  of  life  insurance  policies, 
life  policies  and  endowment  policies. 

(a)  Under  the  ordinary  life  policy,  premiums  are  paid  annually 
during  the  life  of  the  insured. 

{b)  There  is  another  kind  of  life  policy  known  as  the  Limited 
Payment  Life  Policy.  Under  this  policy  the  premiums  are  paid 
during  a  certain  number  of  years  only. 

440.  A  Life  Policy  is  a  contract  on  the  part  of  the 
insurance  company  to  pay  the  beneficiary  a  designated 
sum  upon  the  death  of  the  insured. 

441.  An  Endowment  Policy  is  a  contract  on  the  part 
of  the  insurance  company  to  pay  the  beneficiary  at 
the  death  of  the  insured,  or  after  the  lapse  of  an 
agreed  period  of  time,  if  the  insured  is  then  alive. 

442.  The  following  table  shows  the  rates  charged  by 
an  insurance  company : 


RATIONAL  ARITHMETIC 


ns 


LIFE 

PREMIUMS 

ENDOWMENT  PREMIUMS 

Insurance  of  $1000,  payable  at  death  only 

Insurance  of  $1000,  payable  as 
specified  or  on  prior  decease 

Annual  Premiums 

DURING 

Annual  Payments 

Age 

Age 

Life 

10  Years 

20  Years 

In  15  Years 

In  20  Years 

20 

$18.95 

$43.85 

$27.65 

20 

$68.10 

$49.45 

21 

19.35 

44.55 

28.10 

21 

68.20 

49.55 

22 

19.75 

45.25 

28.55 

22 

68.25 

49.65 

23 

20.20 

46.00 

29.00 

23 

68.35 

49.75 

24 

20.65 

46.75 

29.55 

24 

68.45 

49.85 

25 

21.15 

47.55 

30.05 

25 

68.55 

50.00 

26 

21.65 

48.40 

30.60 

26 

68.70 

50.10 

27 

22.20 

49.25 

31.15 

27 

68.80 

50.25 

28 

22.75 

50.15 

31.75 

28 

68.95 

50.40 

29 

23.35 

51.10 

32.35 

29 

69.10 

50.55 

30 

23.95 

52.05 

33.00 

30 

69.25 

50.75 

31 

24.60 

53.05 

33.65 

31 

69.40 

50.95 

32 

25.30 

54.10 

34.35 

32 

69.55 

51.15 

33 

26.05 

55.20 

35.05 

33 

69.75 

51.35 

34 

26.80 

56.30 

35.80 

34 

69.95 

51.60 

35 

27.65 

57.45 

36.60 

35 

70.20 

51.90 

36 

28.50 

58.65 

37.45 

36 

70.40 

52.15 

37 

29.40 

59.95 

38.30 

37 

70.70 

52.50 

38 

30.35 

61.25 

39.20 

38 

71.00 

52.85 

39 

31.40 

62.60 

40.15 

39 

71.30 

53.25 

40 

32.50 

64.00 

41.20 

40 

71.65 

53.70 

By  this  table  the  premium  on  an  ordinary  life  policy  of  $  1000  at 
the  age  of  20  would  be  $18.95,  for  a  $5000  policy  the  premium 
would  be  5  X  $18.95,  or  $94.75. 

At  the  same  age,  a  $1000  ten-payment  life  policy  would  cost 
$43.85  per  year,  and  a  $5000  policy  5  X  $43.85,  or  $2 19. 25  per  year. 

A  $1000  fifteen-year  endowment  policy  would  cost  $68.10  a  year, 
and  a  $5000  policy  would  cost  $340.50. 

Note.      For  practice  problems  in  life  insurance  see  par.  74. 


EXCHANGE 

443.  Exchange  is  a  system  of  paying  debts  in  distant 
places  by  means  of  drafts. 

444.  A  Draft,  or  Bill  of  Exchange,  is  an  order  of 
one  party  directing  a  second  party  to  pay  a  third 
party  a  certain  sum  of  money  at  a  certain  time. 

445.  There  are  two  kinds  of  bills  of  exchange,  do- 
mestic and  foreign. 

446.  A  Domestic  Bill  of  Exchange,  sometimes  called 
an  inland  hill  of  exchange,  is  one  drawn  and  payable 
in  the  same  state  or  country. 

447.  Foreign  Bills  of  Exchange  are  those  drawn  in 
one  state  or  country  and  payable  in  another  state  or 
country. 

448.  The  Face  or  Par  Value  of  a  bill  of  exchange  is 
the  sum  of  money  for  which  it  is  written. 

Bills  of  exchange  are  always  written  in  the  coinage  of  the  country 
in  which  they  are  to  be  paid. 

DOMESTIC   EXCHANGE 

449.  Domestic  Exchange  quoted  at  a  premium  is 
worth  the  given  percentage  of  the  face  more  than  the 
face. 

124 


RATIONAL  ARITHMETIC  125 

Domestic  Exchange  quoted  at  a  discount  is  worth 
the  quoted  percentage  of  the  face  less  than  the  face. 

(a)  Thus,  ^%  premium  means  that  $1  would  cost  $1,005.  Ex- 
change quoted  at  |%  discount  would  mean  that  $1  would  cost 
$.995. 

(b)  When  Boston  owes  New  York  the  same  that  New  York 
owes  Boston,  exchange  will  be  at  par  in  botli  places.  AMien  Boston 
owes  New  York  more  than  New  York  owes  Boston,  exchange  on 
New  York  will  be  at  a  premium  in  Boston,  since  there  will  be  more 
buyers  of  New  York  exchange  than  sellers ;  and  when  New  York 
owes  Boston  more  money  than  Boston  owes  New  York,  exchange  on 
New  York  will  sell  at  a  discount  in  Boston  and  exchange  on  Boston 
will  sell  at  a  premium  in  New  York. 

The  general  principles  of  percentage  (pars.  242-254) 
are  used  in  solving  problems  in  exchange. 

Note.  For  practice  problems  in  domestic  exchange  see  pars.  75  to  80 
inclusive. 

FOREIGN  EXCHANGE 

450.  Foreign  Exchange  is  exchange  drawn  in  one 
country  and  payable  in  another  country. 

Foreign  bills  are  always  made  in  the  coinage  of  the  country 
where  they  are  to  be  paid. 

451.  The  Intrinsic  Par  of  Exchange  is  the  actual 
value  of  the  money  of  one  country  expressed  in  the 
monev  of  another. 

Intrinsic  value  of  foreign  money  expressed  in  the  money  of  the 
United  States  will  be  found  in  the  table,  par.  468. 

452.  Commercial  Rate  of  Exchange  is  the  market 
value  of  the  money  of  one  country  expressed  in  the 
money  of  another. 


126  RATIONAL  ARITHMETIC 

(a)  This  value  changes  from  time  to  time,  according  to  the 
demand  that  may  exist  and  according  to  the  different  conditions 
of  commerce  that  may  arise. 

(&)  The  rate  of  exchange  on  Great  Britain  is  expressed  by  giv- 
ing the  market  value  of  a  pound  in  United  States  money. 

(c)  On  France,  Belgium,  and  Switzerland  the  rate  of  exchange 
is  expressed  by  giving  the  number  of  francs  that  may  be  secured 

for  $1. 

(d)  On  Germany  the  rate  of  exchange  is  expressed  by  giving 
the  market  value  of  4  reichsmarks  in  United  States  money. 

(e)  On  Holland  the  rate  of  exchange  is  expressed  by  giving  the 
value  of  1  guilder  in  United  States  money. 

(/)  Gold  is  exported  at  a  profit  when  the  cost  of  foreign  ex- 
change is  enough  greater  than  the  intrinsic  value  of  the  bill  to  pay 
the  cost  of  safe  shipment  and  yet  leave  a  margin ;  and  gold  is  im- 
ported at  a  profit  when  the  cost  of  exchange  is  enough  less  than  the 
intrinsic  value  of  the  bill  to  pay  the  same  expenses  and  leave  a 
margin.  Thus  under  normal  conditions,  the  commercial  rate  is  not 
allowed  to  vary  from  the  intrinsic  par  by  more  than  enough  to  pay 
the  expense  of  shipping  gold. 

Note.  For  practice  problems  in  foreign  exchange  see  pars.  81  to  85  in- 
clusive. 


STOCKS  AND  BONDS 

453.  A  Corporation  is  an  association  of  persons 
authorized  by  law  to  act  as  one  person. 

454.  The  Capital  Stock  of  a  corporation  is  the  value 
of  its  investment.  This  is  divided  into  equal  parts 
called  shares. 

455.  The  Par  Value  of  a  share  of  stock  is  the  value 
placed  upon  each  share  at  the  time  of  the  original 
division  of  its  capital  stock. 

The  usual  par  value  of  one  share  of  stock  is  $100.  Stock  is 
frequently  issued  in  other  sized  shares,  however,  usually  $50,  $25, 
$10,  $5,  or  $1. 

456.  A  Stock  Certificate  is  a  document  issued  by 
the  company  to  the  shareholder  specifying  the  number 
and  par  value  of  the  shares  to  which  he  is  entitled. 

457.  The  Market  Value  of  a  share  is  the  sum  for 
which  it  will  sell  in  the  open  market. 

Sometimes  stock  is  worth  more  than  par  and  sometimes  less. 
This  depends  upon  the  condition  of  the  business. 

458.  A  Dividend  is  that  part  of  the  net  earnings  of  a 
corporation  that  is  divided  among  its  stockholders. 

127 


128  RATIONAL  ARITHMETIC 

459.  An  Assessment  is  a  sum  levied  upon  the  stock- 
holders to  make  up  losses. 

There  are  two  kinds  of  stock,  common  and  preferred. 

460.  Common  Stock  participates  in  the  net  earnings 
of  the  company,  after  all  other  expenses  have  been 
met,  in  such  proportion  as  the  directors  of  the  cor- 
poration may  determine. 

461.  Preferred  Stock  participates  in  the  net  earnings 
of  the  corporation  at  a  fixed  rate  before  any  dividend 
may  be  declared  on  the  common  stock. 

462.  A  Bond  is  an  obligation  of  a  corporation  to 
pay  money  on  a  long  term  of  credit. 

(a)  Bonds  are  usually  secured  by  deeds  of  trust  and  mortgages. 
They  are  generally  issued  as  securities  for  loans.  They  are  similar 
to  promissory  notes,  but  are  more  formal  and  are  also  made  under 
seal. 

(6)   Bonds  are  usually  issued  in  $500,  $1000,  or  multiples  thereof. 

(c)    Quotations  on  bonds  are  given  on  $100  par  value. 

{d)  Bonds  are  issued  in  two  classes,  registered  and  coupon. 

463.  Registered  Bonds  are  those  payable  to  the 
order  of  the  owner  and  can  be  transferred  only  by 
acknowledged  assignment. 

Interest  on  registered  bonds  is  paid  by  check  from  the  corpo- 
ration made  to  the  holder  of  record. 

464.  A  Coupon  Bond  is  one  made  payable  to  the 
bearer,  and  has  interest  certificates  attached.  These 
certificates,  called  coupons,  are  to  be  cut  off  as  they 


RATIONAL  ARITHMETIC  129 

become  due  and  presented  at  the  designated  place  for 
payment. 

(a)  Bonds  are  named  from  the  nature  of  the  security ;  the 
name  of  the  corporation  issuing  tliem ;  the  date  on  which  they  are 
payable ;  the  rate  of  interest  they  bear ;  or  the  purpose  for  which 
they  are  issued. 

(h)  Both  stocks  and  bonds  are  quoted  at  some  per  cent  of  par 
value. 

(c)  The  regular  commission  allowed  to  brokers  for  buying  or 
selling  either  stocks  or  bonds  is  |%  of  the  par  value.  There  is  a 
minimum  charge  for  small  transactions,  however. 

(d)  Dividends  and  assessments  are  always  figured  on  the  par 
value  of  the  stock. 

Note.     For  practice  problems  in  stocks  and  bonds  see  par.  86. 


TABLES 

UNITED   STATES   MONEYS 

465.  United  States  Money  consists  of  gold  coins, 
silver  coins,  United  States  Treasury  notes  and  certifi- 
cates, and  national  bank  notes. 

The  unit  of  measure  is  the  gold  dollar  of  25.8  grains. 

10  mills     =lcent(^)  $     .01 

10  cents    =1  dime  (d.)  $     .10 

10  dimes  =1  dollar  ($)  $  1. 

10  dollars  =  1  eagle  (e.)  $10. 

20  dollars  =  1  double  eagle  (d.  e.)  $20. 

ENGLISH   MONEYS 

466.  English  or  Sterling  Money  is  the  legal  currency 
of  Great  Britain. 

The  unit  of  measure  is  the  pound,  worth  $4.8665  in 
United  States  money. 

4  farthings  =  1  penny  (d) 
12  pence  =  1  shilling  (s) 
20  shillings  =  1  pound    (£) 

Note.     21  shillings  =  1  guinea  (used  in  the  retail  trade). 

130 


RATIONAL  ARITHMETIC  131 

FOREIGN   MONEYS 

467.  Once  each  year  the  Director  of  the  United 
States  Mint  is  required  to  compare  the  values  of  foreign 
coins  with  the  United  States  Gold  Dollar  and  certify 
the  result  of  his  comparison  to  the  Secretary  of  the 
Treasury,  who  then  proclaims  the  value  of  foreign 
money  thus  found  to  be  the  value  to  be  used  in  esti- 
mating the  worth  of  all  foreign  merchandise  imported. 
Values  thus  found  are  called  intrinsic  or  real  values 
and  should  be  distinguished  from  commercial  or  ex- 
change values. 

468.  The  table  of  values  on  pages  132  and  133  was 
proclaimed  Oct.  1,  1918. 

WEIGHT 
Troy  Weight 

469.  Troy  Weight  is  used  in  weighing  precious 
metals. 

TABLE 

24  grains  (gr.)      =  1  pennyweight  (dwt.) 
20  penny  weights  =  1  ounce  (oz.) 
12  ounces  =  1  pound  (lb.) 

Diamond  Weight 

470.  Diamond  Weight  is  used  in  weighing  precious 
stones. 

The  unit  is  3^  Troy  grains  and  is  called  a  carat. 

This  carat  is  not  the  same  as  that  used  in  estimating  the  rela- 
tive purity  of  gold  in  coins  and  jewelry.  Pure  gold  is  24  carats 
fine ;   18  carats  fine  means  ^f  pure  gold  and  ^  alloy. 


132 


RATIONAL  ARITHMETIC 


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134  RATIONAL  ARITHMETIC 

Apothecaries'  Weight 

471.  Apothecaries'  Weight  is  used  by  physicians  and 
apothecaries  in  writing  and  preparing  prescriptions  for 
dry  medicines. 

TABLE 

20  grains  (gr.)  =  1  scruple  (sc.  or  3) 
3  scruples       =  1  dram  (dr.  or  3) 
8  drams  =  1  ounce  (oz.  or  §) 

12  ounces  =1  pound  (lb.) 

Avoirdupois  Weight 

472.  Avoir d  'pois  Weight  is  used  in  commerce  in  all 
cases  excepting  those  requiring  Troy  or  Apothecaries' 
weight. 

TABLE 

16  ounces  =1  pound  (lb.) 

25  pounds  =  1  quarter  (qr.) 

4  quarters  =  1  hundredweight  (cwt.) 

20  hundredweights  =  1  ton  (T.) 
2240  pounds  =  1  long  ton 

473.  Comparison  of  Troy  and  Avoirdupois  Weights. 

1  pound  Troy  =  5760  grains 

1  pound  Avoirdupois  =  7000  grains 
1  ounce  Troy  =    427  grains 

1  ounce  Avoirdupois   =   480  grains 

474.  The  following  table  shows  the  weight  of  a 
bushel  used  commercially  in  measuring  grain  and 
other  farm  products : 


RATIONAL  ARITHMETIC 


135 


Barley 

48  1b. 

Oats 

321b 

Beans 

60   " 

Onions 

57   " 

Buckwheat 

48   " 

Peas 

60   " 

Clover  Seed 

60    " 

Potatoes 

60   " 

Corn,  shelled 

56   " 

'I'imothy  Seed 

45   " 

Corn,  in  the  ear 

70   " 

Rye 

56   " 

Corn  Meal 

50   " 

Rye  Meal 

50   " 

Flaxseed 

56   " 

Wheat 

60   " 

Hemp  Seed 

44   " 

Wheat  Bran 

20   " 

Malt 

34   " 

Liquid  Measure 

475.  Liquid  Measure  is  used  in  measuring  liquids. 

TABLE 

4  gills  (gi.)  =  1  pint  (pt.) 
2  pints         =  1  quart  (qt.) 
4  quarts      =  1  gallon  (gal.) 

476.  Standard  liquid  gallon  contains  231  cubic 
inches. 

There  are  various  kinds  of  casks  for  containing 
liquids.  In  commerce  each  is  gauged  and  its  capacity 
marked  upon  it.     The  various  kinds  of  casks  are : 


1  lerce 

about 

42  gal. 

Puncheon 

84    " 

Pipe 

126    " 

Butt 

126    " 

Tun 

252    " 

Hogshead  (hhd.) 

63    " 

136  RATIONAL  ARITHMETIC 

Apothecaries'  Liquid  Measure 

477.  Apothecaries''   Liquid  Measure  is  used  in  pre- 
scribing and  compounding  liquid  medicines. 

TABLE 

60  minims  ("l)    =1  fluid  drachm  (f3) 
8  fluid  drachms  =  1  fluid  ounce  (f  i) 

16  fluid  ounces     =  1  pint  (O) 
8  pints  =  1  gallon  (Cong.) 

The  gallon  of  this  measure  is  the  same  as  the  gallon 
of  the  liquid  measure. 

Dry  Measure 

478.  Dry  Measure  is  used  in  measuring  grain,  fruits, 
vegetables,  etc.,  which  are  not  sold  by  weight. 

TABLE 

2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts         =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bu.) 

Long  Measure 

479.  Long  Measure  is  used  in  measuring  lengths,  or 

distances. 

TABLE 

12  inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 

5i  yards  (16i  ft.)  =  1  rod  (rd.) 

40  rods  =  1  furlong  (fur.) 

8  furlongs  (320  rods)  =  1  mile  (mi.) 


RATIONAL  ARITHMETIC  137 

Surveyors'  Long  Measure 

480.  Surveyors'  Long  Measure  is  used  by  surveyors 
in  measuring  distances. 

TABLE 

7.92  inches  =  1  link  (1.) 
100  links    =  1  chain  (ch.) 
80  chains  =  1  mile  (mi.) 

Square  Measure 

481.  Square  Measure  is  used  in  measuring  extent  of 

surfaces. 

TABLE 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 
9  square  feet  =  1  square  yard  (sq.  yd.) 

30i  square  yards  =  1  square  rod  (sq.  rd.) 

40  square  rods  =  1  rood  (R.) 

4  roods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

Surveyors'  Square  Measure 

482.  Surveyors'  Square  Measure  is  used  by  surveyors 
in  finding  the  area  of  land. 

TABLE 

625  square  links  (sq.  1.)  =  1  square  rod  or  pole  (sq.  rd. 

or  p.) 

16  poles  =1  square  chain  (sq.  ch.) 

10  square  chains  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 


138  RATIONAL  ARITHMETIC 

Cubic  Measure 

483.  Cubic  Measure  is  used  in  measuring  the  con- 
tents of  anything  which  has  length,  breadth,  and 
thickness. 

TABLE 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

Wood  Measure 

484.  Wood  Measure  is  used  in  measuring  wood. 

TABLE 

16  cubic  feet  =  1  cord  foot  (cd.  ft.) 

8  cord  feet  (128  cu.  ft.)  =  1  cord  (cd.) 

A  cord  of  wood  is  a  pile  8  feet  long,  4  feet  wide,  and 
4  feet  high,  or  its  equivalent. 


485. 


TIME 

60  seconds  (sec.)  =  1  minute  (min.) 

60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da.) 

7  days  =  1  week  (wk.) 

30  days  =  1  month  (mo.) 

52  weeks  =  1  year  (yr.) 

12  months  =  1  year  (yr.) 

365  days  =  1  common  year 

366  days  =  1  leap  year 
100  years  =  1  century 


RATIONAL  ARITHMETIC  139 

486.  The  day  is  the  time  during  which  the  earth 
makes  one  revolution  on  its  own  axis. 

487.  The  Solar  Year  is  the  time  the  earth  requires 
to  make  one  complete  revolution  around  the  sun.  It 
actually  takes  the  earth  365^  days  to  make  this  revolu- 
tion. Therefore,  every  fourth  year  is  given  366  days. 
This  extra  day  is  added  to  the  month  of  February,  the 
shortest  month,  and  the  year  is  called  Leap  Year. 

These  figures  are  not  absolutely  accurate  but  are  practically 
so. 

488.  Any  year  whose  number  can  be  divided  by  4 
is  a  leap  year,  except  that  a  century  year  must  be 
divisible  by  400. 

(a)  The  year  1916  could  be  divided  by  4  and  was  a  leap  year, 
while  1915  could  not  be  divided  bv  4  and  was  an  ordinarv  year. 

(b)  The  year  1900  was  divisible  by  4  but  not  by  400  and  was 
not,  therefore,  a  leap  year.  The  year  2000  being  divisible  by  4 
and  400  will  be  a  leap  year. 

489.  Months  of  the  Year,  and  Days  in  Each  : 


1. 

Januarv 

31 

7. 

July 

31 

2. 

February 

28  or  29 

8. 

August 

31 

3. 

March 

31 

9. 

September 

30 

4. 

April 

30 

10. 

October 

31 

5. 

Mav 

31 

11. 

November 

30 

6. 

June 

30 

12. 

December 

31 

140  RATIONAL  ARITHMETIC 

MISCELLANEOUS 

490.  Some  articles  are  sold  by  quantity  according 
to  the  following  table  : 

TABLE 

12  units  =  1  dozen  (doz.) 

12  dozen  =  1  gross  (gr.) 

12  gross  =  1  great  gross  (g.  gr.) 

20  units  =  1  score 

Paper  Measure 

491.  Paper  is  measured  according  to  the  following 
table : 

TABLE 

24  sheets     =  1  quire  (qu.) 
20  quires     =  1  ream  (rm.) 

2  reams     =  1  bundle  (bdl.) 

5  bundles  =  1  bale  (bl.) 


THE   METRIC   SYSTEM 

492.  The  Metric  System  is  a  decimal  system  of 
weights  and  measures,  similar  to  the  decimal  system 
used  in  measuring  United  States  money.  It  was 
originated  in  France  early  in  the  nineteenth  century, 
and  has  been  adopted  by  nearly  all  the  commercial 
nations  except  United  States  and  England. 

The  Metric  System  was  made  legal  in  the  United 
States  in  1866,  but  is  not  generally  used  except  in 
scientific  work. 


RATIONAL  ARITHMETIC  141 

493.  The  Meter.  The  basic  unit  is  the  meter. 
The  other  units,  those  of  weight  and  of  capacity,  are 
based  on  it. 

494.  The  length  of  the  meter  was  originally  deter- 
mined by  taking  one  ten-millionth  of  the  distance 
from  the  equator  to  the  pole.  This  length  is  39.37 
inches. 


495.    The  primary  units  are  : 

For  length  —  meter 
For  capacity  —  liter 
For  weight  —  gram 


496.  The    desired    integral    multiples    of    these    are 
formed  by  using  the  following  Greek  prefixes : 

Deca    =10  (decameter  =  10  meters) 
Hecto  =  100  (hectometer  =  100  meters) 
Kilo     =  1000  (kilometer  =  1000  meters) 
Myria=  10,000  (myriameter=  10,000  meters) 

497.  To  designate  decimals  of  a  meter,  the  following 
Latin  prefixes  are  used  : 

Deci  =  To       (decimeter  =  ro  meter) 
Centi  =  -TWO     (centimeter  =  two  meter) 
Milli  =  ToVo  (millimeter  =  x^oo  meter) 

The    most    commonly    used    denominations    in    the 
following  tables  are  indicated  by  heavy-faced  type. 


142  RATIONAL  ARITHMETIC 

Linear  Measure 

498.  The  unit  of  Linear  Measure  is  the  meter. 

TABLE 

10  millimeters  (mm.)  =  1  centimeter  (cm.) 

10  centimeters  =  1  decimeter  (dm.) 

10  decimeters  =  1  meter  (m.) 

10  meters  =  1  decameter  (dm.) 

10  decameters  =  1  hectometer  (hm.) 

10  hectometers  =  1  kilometer  (km.) 

10  kilometers  =  1  myriameter  (mm.) 

Square  Measure 

499.  The  unit  of  Square  Measure  is  the  square  meter. 

TABLE 

100  square  milHmeters  =  1  square  centimeter  (cmq.) 
100  square  centimeters  =  1  square  decimeter  (dmq.) 
100  square  decimeters    =  1  square  meter  (mq.) 
100  square  meters  =  1  square  decameter  (dcmq.) 

100  square  decameters   =  1  square  hectometer  (sq.  hm.) 
100  square  hectometers  =  1  square  kilometer  (sq.  km.) 

Land  Measure 

500.  The  unit  of  Land  Measure  is  the  are. 

TABLE 

100  centiares  (ca.)  =  1  are  (a.)  =  100  sq.  m. 

100  ares  =  1  hectare  (ha.)  =  10,000  mq. 


RATIONAL  ARITHMETIC  143 

Cubic  Measure 

501.  The  unit  of  volume  is  the  cubic  meter. 

TABLE 

100  cubic  millimeters  (cmm.)  =  1  cubic  centimeter  (cmc.) 
100  cubic  centimeters  =  1  cubic  decimeter  (dmc.) 

100  cubic  decimeters  =  1  cubic  meter  (mc.) 

Wood  Measure 

502.  The  unit  of  wood  measure  is  the  stere. 

TABLE 

10  decisteres  (dst.)  =  1  stere  (st.)  =  1  cu.  m. 

10  steres  =  1  decastere  (dast.)  =  10  cu.  m. 

Measure  of  Capacity 

503.  The  unit  of  capacity  for  either  solids  or  liquids 
is  the  liter,  which  is  equal  in  volume  to  1  cu.  dm. 

TABLE 

10  milliliters  (ml.)  =  1  centiliter  (cl.) 
10  centiliters  =  1  deciliter  (dl.) 

10  deciliters  =  1  liter  (1.) 

10  liters  =  1  decaliter  (dl.) 

10  decaliters  =  1  hectoliter  (hi.) 

10  hectoliters         =  1  kiloKter  (kl.) 

Measure  of  Weight 

504.  The  unit  of  weight  is  the  gram,  which  is  the 
weight  of  1  dmc.  of  distilled  water  in  a  vacuum,  at  its 
greatest  density.     It  weighs  15.4324  gr. 


144 


RATIONAL  ARITHMETIC 


TABLE 

10  milligrams  (mg.)  = 

10  centigrams  = 

10  decigrams  = 

10  grams  = 

10  decagrams  = 

10  hectograms  = 

10  kilograms  = 

10  myriagrams  = 

10  quintals  = 


centigram  (eg.) 
decigram  (dg.) 
gram  (g.) 
decagram  (dg.) 
hectogram  (hg.^ 
kilogram  (kg.) 
myriagram  (mg.) 
quintal  (q.) 
tonneau  (t.) 


505. 


TABLES   OF   EQUIVALENTS 

Convenient  Equivalent  Values 

1  cu.  cm.  of  water  =  1   ml.   of  water,  and  weighs   1   gram 

=  15.432  gr. 

1  cu.  dm.  of  water  =  1  1.  of  water,  and  weighs  1  kg. 

=  2.2046  lb. 

1  cu.  m.  of  water  =1  kl.  of  water,  and  weighs  1  tonneau 

=  2204.6  lb. 


506. 


Measures  of  Weight 

1  grain,  Troy  =  .0648  of  a  gram 

1  ounce,  Troy  =31.104  grams 

1  ounce,  Avoir.  =28.35  grams 

1  lb.  Troy  =  .3732  of  a  kilogram 

1  lb.  Avoir.  =  .4536  of  a  kilogram 

1  ton  (short)  =  .9072  of  a  tonneau  or  ton 


RATIONAL  ARITHMETIC 


145 


1  gram 
1  gram 
1  gram 
1  kilogram 
1  kilogram 
1  tonneau 


=  15.432  grains,  Troy 
=  .03215  of  an  oz.  Troy 
=  .03527  of  an  oz.  Avoir. 
=  2.679  lb.  Troy 
=  2.2046  lb.  Avoir. 
=  1.1023  tons  (short) 


507. 


Measures  of  Capacity 
1  dry  quart  =1.101  liters 


1  liquid  quart 
1  liquid  gallon 
1  peck 
1  bushel 


=  .9463  of  a  liter 
=  .3785  of  a  decaliter 
=  .881  of  a  decaliter 
=  .3524  of  a  hectoliter 


1  liter 
1  liter 
1  decaliter 
1  decaliter 
1  hectoliter 


=  .908  of  a  dry  quart 
=  1.0567  liquid  quarts 
=  2.6417  liquid  gal. 
=  1.135  pecks 
=  2.8377  bushels 


508. 


1  inch 
1  foot 
1  yard 
1  rod 
1  mile 


Linear  Measure 

=  2.54  centimeters 
=  .3048  of  a  meter 
=  .9144  of  a  meter 
=  5.029  meters 
=  1.6093  kilometers 


146  RATIONAL  ARITHMETIC 

1  centimeter  =  .3937  of  an  inch 

1  decimeter  =  .328  of  a  foot 

1  meter  =1.0936  yards 

1  dekameter  =1.9884  rods 

1  kilometer  =  .62137  of  a  mile 


509. 


Surface  Measure 

sq.  inch  =6.452  sq.  centimeters 

sq.  foot  =  .0929  of  a  sq.  meter 

sq.  yard  =  .8361  of  a  sq.  meter 

sq.  rod  =25.293  sq.  meters 

acre  =40.47  ares 

sq.  mile  =259  hectares 

sq.  centimeter  =  .155  of  a  sq.  inch 

sq.  decimeter  =.1076  of  a  sq.  foot 

sq.  meter  =1.196  sq.  yards 

are  =3.954  sq.  rods 

hectare  =2.471  acres 

sq.  kilometer  =  .3861  of  a  sq.  mile 


510. 


Cubic  Measure 

1  cu.  inch  =  16.387  cu.  centimeters 

1  cu.  foot  =28.317  cu.  decimeters 

1  cu.  yard  =  .7646  of  a  cu.  meter 

1  cord  =3.624  steres 

1  cu.  centimeter  =  .061  of  a  cu.  inch 
1  cu.  decimeter  =  .0353  of  a  cu.  foot 
1  cu.  meter  =1.308  cu.  vards 

1  stere  =  275)  of  a  cord 


INDEX 


(Figures  refer  to  paragraph  numbers.) 


Accident   insurance,   437. 

Account  purchase,  definition,  309. 

Account  purchases,  problems  in 
making,  48. 

Account  sales,  definition,  308 ;  prob- 
lems in  making,  48. 

Accurate  interest,  definition,  325 ; 
illustrated  solutions,  327,  328,  341, 
343;    problems  in  finding,  56. 

Addend,  definition,  97. 

Addition,  compound  numbers,  233; 
decimals,  120,  131 ;  definition, 
88;  fractions,  18,  201,  205;  in- 
tegers, 1,  92,  99;   proof  of,  100. 

Ad  valorem  duty,  406. 

Aliquot  parts,  definitions,  240;  illus- 
trated solution,  241 ;  problems, 
27,  30;    table  of,  241. 

Amount,  250,  322,  369. 

Amount  of  purchase,  to  find,  illus- 
trated solution,  311. 

Annual  interest,  389;  illustrated 
solution,  390. 

Apothecaries'  liquid  measure,  477. 

Apothecaries'  weight,  471. 

Appraiser,  410. 

Arithmetic,  definition,  87. 

Asking  price,  problems  in  finding,  45. 

Assessment,  459. 

Average  clause,  432. 

Averaging  accounts,  definition,  391 ; 
general  principles  of.  393,  395 ; 
illustrated  solutions,  397,  400; 
problems,   68;    rule,   396. 

Avoirdupois  weight,  472. 


Balancing  accounts,  109 ;     problems, 

9. 
Bank  discount,  definition,  376,  379; 

illustrated  solution,  382,  385 ;    to 

find  face,  64 ;   to  find  proceeds,  63. 
Base    definition,    247;     problems    in 

finding,    34;     to    find,    illustrated 

solution,  259,  261. 
Bill  of  exchange,  444. 
Billing,  exercises  in,   31 ;    problems, 

46. 
Bond,  462 ;   coupon,  464 ;   registered, 

463. 
Bonded  warehouse,  420. 
Broker,  definition,  300. 
Brokerage,  see  Commission. 
Butt,  476. 

Capital  stock,  454. 

Cash  balance,  392 ;  problems  in  find- 
ing, 68. 

Casks,  476. 

Charges,  definition,  304. 

Check,  definition,  370. 

Collector  of  customs,  414. 

Commercial  paper,  definition,  357. 

Commercial  rate  of  exchange,  452. 

Commission,  definitions,  302,  310; 
to  find,  illustrated  solution,  310, 
311. 

Commission  and  brokerage,  illus- 
trated solutions,  311,  316. 

Commission  merchant,  definition, 
299. 

Commission  problems,  to  find  com- 


147 


148 


INDEX 


mission,  47 ;   to  find  gross  cost,  47 ; 

to  find  gross  proceeds,  49;    to  find 

net  proceeds,  47 ;   to  find  rate,  50 ; 

general  problems,  51. 
Common  divisor,  definition,  153. 
Common    fraction,    definition,    165 ; 

changing  to,  193. 
Common  multiple,  definition,  148. 
Common  stock,  460. 
Complex  decimal,  definition,  173. 
Complex  fractions,  definition,  174. 
Composite  numbers,  definition,  144. 
Compound  interest,  definition,  386; 

illustrated  solution,  388 ;  problems 

in  finding,  66 ;   rule,  387. 
Compound  subtraction,  problems  in, 

52. 
Corporation,  453. 
Cost,  definition,  266,  268;    problems 

in  finding,  38 ;    to  find,  illustrated 

solution,  276,  278. 
Coupon  bond,  464. 
Cubic  measure,  483. 
Customhouse  business,  410,  416. 
Customs,  405. 
Customs  and  duties,  problems,  72. 

Date  of  maturity,  definition,  367. 

Decimal  fraction,  definition,  166. 

Decimal  fractions,  changing  to,  196. 

Decimals,  10;  division  problems,  11 ; 
multiplication  problems,  10;  com- 
plex,  173;    mixed,   171. 

Denominate  numbers,  addition,  233; 
change  to  higher  denomination, 
230;  change  to  lower  denomina- 
tion, 231 ;  changing  to  simple, 
232 ;  definitions,  226,  228 ;  divi- 
sion, 236 ;  multiplication,  235 ; 
reduction,  illustrated  solution,  229 ; 
reduction  problems,  23,  24,  25,  26 ; 
subtraction,  234. 

Denominator,  definition,  192. 

Diamond  weight,  470. 


Difference,  definition,  251,  105. 

Discount,  288;  definitions,  281,  283; 
series,  289;  to  find,  illustrated 
solution,  291,  292. 

Dividend,  458. 

Dividend,  definition,  120. 

Division,  compound  numbers,  236; 
decimals,  11,  136;  decimals,  illus- 
trated solutions,  139,  141 ;  deci- 
mals, rule,  137 ;  definition,  91 ; 
fractions,  219 ;  fractions,  problems, 
21;  integers,  definition,  119;  in- 
tegers, illustrated  solutions,  124, 
126;    integers,  proof,  125,  127. 

Divisor,  definition,  121. 

Domestic  bill  of  exchange,  446. 

Domestic  exchange,  449. 

Draft,  444;   definition,  361. 

Drawee,  definition,  363. 

Drawer,  definition,  362. 

Dry  measure,  478. 

Duties,  405. 


Endowment  policy,  441. 

English  money,  466. 

Exact  days,  problems  in  finding,  53. 

Exchange,  definition,  443 ;  domestic, 
to  find  face  value  of  a  draft,  78,  79 
to  find  value  of  sight  draft,  75 
to  find  value  of  time  draft,  76 
foreign,  450;  to  find  value  of 
draft,  81. 

Excise  duty,  421. 

Face,  448. 

Factor,  definition,  114. 

Factoring,  142. 

Fire  insurance,  430 ;   problems,  73. 

Foreign  bill  of  exchange,  447. 

Foreign  exchange,  450. 

Fraction,  definition,  160. 

Fractions,   addition,   201 ;    addition, 

illustrated     solution,     203,     205; 

addition,  problems,   18;    addition. 


INDEX 


149 


rule,  202;  changing  to  common, 
illustrated  solution,  195;  chang- 
ing to  common,  rule,  190;  illus- 
trated solution,  198,  200 ;  changing 
to  a  decimal,  rule,  197;  division 
problems,  21;  division,  rule,  221, 
223;  changing  to  higher  terms, 
illustrated  solutions,  186 ;  chang- 
ing to  higher  terms,  rule,  185 ; 
changing  to  improper  fractions, 
rule,  191 ;  change  to  lower  terms, 
illustrated  solutions,  181,  183; 
change  to  lower  terms,  rule,  180; 
changing  to  mixed  numbers,  illus- 
trated solution,  189;  changing 
to  mixed  numbers,  rule,  188 ;  com- 
plex, 174 ;  general  problems,  22 ; 
multiplication,  illustrated  solution, 
211,  212,  214,  216,  218;  multipli- 
cation problems,  20;  multiplica- 
tion, rules,  210,  213,  215;  prob- 
lems in  reduction,  12,  22;  sub- 
traction, 206 ;  subtraction,  illus- 
trated solution,  207. 
Free  list,  409. 

Gain,  see  Profit. 

Gram,  495,  504. 

Greatest  common  divisor,  definition, 
154 ;  illustrated  solution,  156,  158, 
159. 

Gross  amount,  definition,  286 ;  prob- 
lems in  finding,  43;  to  find,  illus- 
trated solution,  294,  296. 

Gross  cost,  definition,  268,  307. 

Gross  proceeds,  definition,  303; 
problems  in  finding,  49. 

Gross  sales,  to  find,  illustrated  solu- 
tion, 312,  314. 

Gross  selling  price,  definition,  269. 

Health  insurance,  438. 

Higher  terms,  changing  to,  184. 

Hogshead,  476. 


Improper  fraction,  190;  definition, 
169. 

Income  tax,  404. 

Insurance,  definition,  423 ;  problems, 
73;    table,  442. 

Interest,  combinations  of  time,  344 ; 
definition,  319;  problems,  com- 
pound interest,  66;  general  prob- 
lems, 60 ;  periodic  interest,  67 ; 
to  find  accurate  interest,  5Q;  to 
find  interest,  54,  55,  56;  to  find 
principal,  59;  to  find  rate,  58; 
to  find  time,  57 ;  rates  other  than 
six,  345;  to  find  interest  on  $1, 
rule,  355 ;  illustrated  solution,  356 ; 
to  find  principal,  illustrated  solu- 
tions, 352,  354 ;  to  find  principal, 
rules,  351,  352;  to  find  rate,  illus- 
trated solutions,  350 ;  to  find  rate, 
rule,  349;  to  find  the  time,  illus- 
trated solutions,  347,  348 ;  to  find 
the  time,  rule,  346. 

Internal  revenue,  417. 

Intrinsic  par,  451. 

Invoice,  419. 

Least  common  multiple,  definition, 
149;    illustrated  solution,  151. 

Legal  rate,  definition,  320. 

Life  insurance,   436;    problems,   74. 

Life  policy,  440. 

Liquid  measure,  475. 

List  price,  definition,  285 ;  to  find, 
illustrated  solution,  294,  296. 

Liter,  495,  503. 

Long  measure,  479. 

Loss,  definition,  272;   to  find,  274. 

Lower  terms,  changing  to,  180. 

Maker,  definition,  359. 
Manifest,  418. 
Marine  insurance,  434. 
Market  value,  457. 


150 


INDEX 


Marking  price,  to  find,  illustrated 
solution,  298. 

Maturity,  date  of,  367;  definition, 
377. 

Merchants'  rule,  partial  payment, 
374;    problems,  62. 

Meter,  493. 

Metric  equivalents,  505,  509. 

Metric  system,  492,  510. 

Minuend,  definition,  103. 

Miscellaneous  measures,  490. 

Mixed  decimal,  definition,  171. 

Mixed  numbers,  changing  to,  188, 
189;   definition,  170. 

Money,  English,  466;  foreign,  467; 
table  of,  468 ;    United  States,  465. 

Multiplicand,  definition.  111. 

Multiplication,  compound  numbers, 
235 ;  problems  in  decimals,  10 ; 
problems  in  fractions,  20;  deci- 
mals, 133;  decimals,  illustrated 
solution,  135;  decimals,  rule,  134; 
definition,  90 ;  fractions,  208 ; 
integers,  definition,  110;  integers, 
illustrated  solutions,  115,  117; 
proof,  116,  118. 

Multiplier,  definition,  112. 

Mutual  Insurance  company,  427. 

Net  amount,  definition,  287 ;  to  find, 
illustrated  solution,  291,  292. 

Net  proceeds,  definition,  305 ;  prob- 
lems in  finding,  47,  48;  to  find, 
illustrated  solution,  313. 

Net  selling  price,  definition,  270. 

Notation,  92. 

Note,  definition,  358. 

Numerator,  definition,  162. 

Ordinary  interest,  definition,  329; 
rule,  332 ;  sixty-day  method,  331 ; 
sixty-day  rule,  illustrated  solu- 
tions, 3.S2,  336. 


Paper  measure,  491. 

Par  value,  448,  455. 

Partial  payments,  merchants'  rule, 
374;  problems  in  merchants' 
rule,  62,  375;  problems  in  United 
States  rule,  61,  373;  problems, 
371,  374;  liuited  States  rule, 
372. 

Payee,  364. 

Pavee,  definition,  360. 

Percentage,  definitions,  243,  254; 
general  problems,  36;  to  find,  33, 
255,  258;  to  find  base,  34,  259, 
261 ;   to  find  rate,  35,  262,  263. 

Periodic  interest,  389;  illustrated 
solution,  390. 

Pipe,  476. 

Policy,  428. 

Poll  tax,  402. 

Port  of  delivery,  412. 

Port  of  entry,  411. 

Preferred  stock,  461. 

Premium,  429. 

Prime  cost,  definition,  267,  306. 

Prime  factors,  definition,  145 ;  illus- 
trated solution,  146. 

Prime  numbers,  definition,  143. 

Principal,  definition,  301,  318;  prob- 
lems in  finding,  59. 

Proceeds,  380;  definition,  303,  305; 
problems  in  finding,  63,  64 ;  to 
find,  381. 

Product,  definition,  113. 

Profit,  definition,  271;  to  find, 
illustrated  solution,  274. 

Profit  and  loss,  general  problems, 
37,  38,  39,  40;  illustrated  solu- 
tions, 274,  280;  problems,  42; 
to  find  cost,  38;  to  find  profit  or 
loss,  37;  to  find  rate  of  profit  or 
loss,  39,  279,  280. 

Profits  and  losses,  definitions,  264, 
272. 

Proper  fraction,  definition,  168. 


INDEX 


151 


Property  tax,  403. 
Puncheon,  476. 

Quotient,  definition,  122. 

Rate,     definitions,    248,    319,    320; 

problems  in  finding,  35,  39,  44,  50, 

58;    to  find,  illustrated  solutions, 

262,  263,  297. 
Rate  of  discount,  to  find  single  rate 

equal    series,    illustrated   solution, 

291. 
Reduction,      denominate     numbers, 

problems,  23,  24,  25,  26 ;  fractions, 

12,  13,  14,  15,  16,  17. 
Reduction    of    fractions,    definition, 

176. 
Registered  bond,  463. 
Remainder,  definition,  105. 

Selling  price,  definition,  269,  270; 
to  find,  illustrated  solution,  274, 
275. 

Short  rate,  433. 

Sight  draft,  365. 

Sixty-day  method,  ordmars''  interest, 
illustrated  solution,  333,  336; 
ordinary  interest,  rule,  332. 

Specific  duty,  407. 

Square  measure,  481. 

Stock,  454;  certificate,  456;  com- 
mon, 460;  company,  426;  pre- 
ferred, 461. 

Stocks  and  bonds,  453,  464 ;  general 
problems,  86. 

Subtraction,  compound  numbers, 
234;  decimals,  120;  definition, 
89;  fractions,  207;  fractions, 
problems,  19;  illustrated  solu- 
tion, 132;    integers,  9,  102;    illus- 


trated solution,  106,   109;    proof, 

107. 
Subtrahend,  definition,  104. 
Sum,  definition,  98. 
Surveyor,  customs,  415. 
Surveyor's  long  measure,  480 ;  square 

measure,  482. 

Tariff,  408. 

Taxes,  401 ;   problems,  71. 

Term  of  discount,  378. 

Tierce,  476. 

Time,  485 ;  definition,  321 ;  method? 
of  computing,  238,  239;  problems 
in  finding,  52,  53. 

Time  discount,  definition,  283. 

Time  draft,  365. 

Trade  discount,  definition,  281,  284^ 
289;  general  problems,  46;  illus- 
trated solutions,  291,  298;  tc 
find  asking  price,  45;  to  find 
gross  amount,  43;  to  find  net 
amount,  50 ;  to  find  rate,  44 ; 
to  find  single  discount  equal  to  a 
series,  45. 

Troy  weight,  469. 

Tun,  476. 

United  States  money,  465. 

United  States  rule,  partial  payment, 
illustrated  solution,  373 ;  problems 
in  partial  payments,  61 ;  partial 
payments,  rule,  372. 

Warehouse,  bonded,  420. 

Weight,  apothecaries',  471 ;  avoir- 
dupois, 472 ;  comparison,  473  : 
diamond,  470;    troy,  469. 

Weights,  miscellaneous,  490. 

Wood  measure,  484. 


ye  35873 


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